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Semiring

In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices.

The smallest semiring that is not a ring is the two-element Boolean algebra, e.g. with logical disjunction $$\lor$$ as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers $$\N$$ under ordinary addition and multiplication, when including the number zero. Semirings are abundant, because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid.

Terminology
Some authors call semiring the structure without the requirement for there to be a $$0$$ or $$1$$. This makes the analogy between ring and on the one hand and  and  on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggesting that rigs are rings without negative elements. (And this is similar to using rng to mean a ring without a multiplicative identity.)

The term dioid (for "double monoid") has been used to mean semirings or other structures. It was used by Kuntzmann in 1972 to denote a semiring. (It is alternatively sometimes used for naturally ordered semirings but the term was also used for idempotent subgroups by Baccelli et al. in 1992.)

Definition
A semiring is a set $$R$$ equipped with two binary operations $$+$$ and $$\cdot,$$ called addition and multiplication, such that:

Explicitly stated, $$(R, +)$$ is a commutative monoid. Further, the following axioms tie to both operations:
 * $$(R, +)$$ is a monoid with identity element called $$0$$:
 * $$(a + b) + c = a + (b + c)$$
 * $$0 + a = a$$
 * $$a + 0 = a$$
 * $$(R, \,\cdot\,)$$ is a monoid with identity element called $$1$$:
 * $$(a \cdot b) \cdot c = a \cdot (b \cdot c)$$
 * $$1 \cdot a = a$$
 * $$a \cdot 1 = a$$
 * Addition is commutative:
 * $$a + b = b + a$$
 * Through multiplication, any element is left- and right-annihilated by the additive identity:
 * $$0 \cdot a = 0$$
 * $$a \cdot 0 = 0$$
 * Multiplication left- and right-distributes over addition:
 * $$a \cdot (b + c) = (a \cdot b) + (a \cdot c)$$
 * $$(b + c) \cdot a = (b \cdot a) + (c \cdot a)$$

Notation
The symbol $$\cdot$$ is usually omitted from the notation; that is, $$a \cdot b$$ is just written $$ab.$$

Similarly, an order of operations is conventional, in which $$\cdot$$ is applied before $$+$$. That is, $$a + b\cdot c$$ denotes $$a + (b\cdot c)$$.

For the purpose of disambiguation, one may write $$0_R$$ or $$1_R$$ to emphasize which structure the units at hand belong to.

If $$x\in R$$ is an element of a semiring and $$n\in{\mathbb N}$$, then $$n$$-times repeated multiplication of $$x$$ with itself is denoted $$x^n$$, and one similarly writes $$x\,n:=x+x+\cdots+x$$ for the $$n$$-times repeated addition.

Construction of new semirings
The zero ring with underlying set $$\{0\}$$ is also a semiring, called the trivial semiring. This triviality can be characterized via $$0=1$$ and so $$0\neq 1$$ is often silently assumed as if it were an additional axiom. Now given any semiring, there are several ways to define new ones.

As noted, the natural numbers $${\mathbb N}$$ with its arithmetic structure form a semiring. Taking the zero and the image of the successor operation in a semiring $$R$$, i.e. the set $$\{x\in R\mid x=0_R\lor \exists p. x = p + 1_R\}$$ together with the inherited operations, is always a sub-semiring of $$R$$.

If $$(M, +)$$ is a commutative monoid, function composition provides the multiplication to form a semiring: The set $$\operatorname{End}(M)$$ of endomorphisms $$M \to M$$ forms a semiring, where addition is defined from pointwise addition in $$M$$. The zero morphism and the identity are the respective neutral elements. If $$M = R^n$$ with $$R$$ a semiring, we obtain a semiring that can be associated with the square $$n\times n$$ matrices $${\mathcal M}_n(R)$$ with coefficients in $$R$$, the matrix semiring using ordinary addition and multiplication rules of matrices. Yet more abstractly, given $$n\in{\mathbb N}$$ and $$R$$ a semiring, $${\mathcal M}_n(R)$$ is always a semiring also. It is generally non-commutative even if $$R$$ was commutative.

Dorroh extensions: If $$R$$ is a semiring, then $$R\times{\mathbb N}$$ with pointwise addition and multiplication given by $$\langle x,n\rangle\bullet \langle y,m\rangle:=\langle x\cdot y+(x\,m+y\,n), n\cdot m\rangle$$ defines another semiring with multiplicative unit $$1_{R\times{\mathbb N}}:=\langle 0_R,1_{\mathbb N}\rangle$$. Very similarly, if $$N$$ is any sub-semiring of $$R$$, one may also define a semiring on $$R\times N$$, just by replacing the repeated addition in the formula by multiplication. Indeed, these constructions even work under looser conditions, as the structure $$R$$ is not actually required to have a multiplicative unit.

Zerosumfree semirings are in a sense furthest away from being rings. Given a semiring, one may adjoin a new zero $$0'$$ to the underlying set and thus obtain such a zerosumfree semiring that also lacks zero divisors. In particular, now $$0\cdot 0'=0'$$ and the old semiring is actually not a sub-semiring. One may then go on and adjoin new elements "on top" one at a time, while always respecting the zero. These two strategies also work under looser conditions. Sometimes the notations $$-\infty$$ resp. $$+\infty$$ are used when performing these constructions.

Adjoining a new zero to the trivial semiring, in this way, results in another semiring which may be expressed in terms of the logical connectives of disjunction and conjunction: $$\langle\{0,1\},+,\cdot,\langle 0,1\rangle\rangle=\langle\{\bot,\top\},\lor,\land,\langle\bot,\top\rangle\rangle$$. Consequently, this is the smallest semiring that is not a ring. Explicitly, it violates the ring axioms as $$\top\lor P = \top$$ for all $$P$$, i.e. $$1$$ has no additive inverse. In the self-dual definition, the fault is with $$\bot\land P = \bot$$. (This is not to be conflated with the ring $$\Z_2$$, whose addition functions as xor $$\veebar$$.) In the von Neumann model of the naturals, $$0_\omega:=\{\}$$, $$1_\omega:=\{0_\omega\}$$ and $$2_\omega:=\{0_\omega,1_\omega\}={\mathcal P}1_\omega$$. The two-element semiring may be presented in terms of the set theoretic union and intersection as $$\langle {\mathcal P}1_\omega,\cup,\cap,\langle \{\},1_\omega\rangle\rangle$$. Now this structure in fact still constitutes a semiring when $$1_\omega$$ is replaced by any inhabited set whatsoever.

The ideals on a semiring $$R$$, with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring. The ideals of $${\mathcal M}_n(R)$$ are in bijection with the ideals of $$R$$. The collection of left ideals of $$R$$ (and likewise the right ideals) also have much of that algebraic structure, except that then $$R$$ does not function as a two-sided multiplicative identity.

If $$R$$ is a semiring and $$A$$ is an inhabited set, $$A^*$$ denotes the free monoid and the formal polynomials $$R[A^*]$$ over its words form another semiring. For small sets, the generating elements are conventionally used to denote the polynomial semiring. For example, in case of a singleton $$A=\{X\}$$ such that $$A^*=\{\varepsilon,X,X^2,X^3,\dots\}$$, one writes $$R[X]$$. Zerosumfree sub-semirings of $$R$$ can be used to determine sub-semirings of $$R[A^*]$$.

Given a set $$A$$, not necessarily just a singleton, adjoining a default element to the set underlying a semiring $$R$$ one may define the semiring of partial functions from $$A$$ to $$R$$.

Given a derivation $${\mathrm d}$$ on a semiring $$R$$, another the operation "$$\bullet$$" fulfilling $$X\bullet y=y\bullet X+{\mathrm d}(y)$$ can be defined as part of a new multiplication on $$R[X]$$, resulting in another semiring.

The above is by no means an exhaustive list of systematic constructions.

Derivations
Derivations on a semiring $$R$$ are the maps $${\mathrm d}\colon R\to R$$ with $${\mathrm d}(x+y)={\mathrm d}(x)+{\mathrm d}(y)$$ and $${\mathrm d}(x\cdot y)={\mathrm d}(x)\cdot y+x\cdot {\mathrm d}(y)$$.

For example, if $$E$$ is the $$2\times 2$$ unit matrix and $$U=\bigl(\begin{smallmatrix}0 & 1 \\ 0 & 0 \end{smallmatrix}\bigr)$$, then the subset of $${\mathcal M}_2(R)$$ given by the matrices $$a\,E + b\,U$$ with $$a,b\in R$$ is a semiring with derivation $$a\,E + b\,U\mapsto b\,U$$.

Properties
A basic property of semirings is that $$1$$ is not a left or right zero divisor, and that $$1$$ but also $$0$$ squares to itself, i.e. these have $$u^2=u$$.

Some notable properties are inherited from the monoid structures: The monoid axioms demand unit existence, and so the set underlying a semiring cannot be empty. Also, the 2-ary predicate $$x\le_\text{pre}y$$ defined as $$\exists d. x + d = y$$, here defined for the addition operation, always constitutes the right canonical preorder relation. Reflexivity $$y\le_\text{pre} y$$ is witnessed by the identity. Further, $$0\le_\text{pre}y$$ is always valid, and so zero is the least element with respect to this preorder. Considering it for the commutative addition in particular, the distinction of "right" may be disregarded. In the non-negative integers $$\N$$, for example, this relation is anti-symmetric and strongly connected, and thus in fact a (non-strict) total order.

Below, more conditional properties are discussed.

Semifields
Any field is also a semifield, which in turn is a semiring in which also multiplicative inverses exist.

Rings
Any field is also a ring, which in turn is a semiring in which also additive inverses exist. Note that a semiring omits such a requirement, i.e., it requires only a commutative monoid, not a commutative group. The extra requirement for a ring itself already implies the existence of a multiplicative zero. This contrast is also why for the theory of semirings, the multiplicative zero must be specified explicitly.

Here $$-1$$, the additive inverse of $$1$$, squares to $$1$$. As additive differences $$d=y-x$$ always exist in a ring, $$x\le_\text{pre}y$$ is a trivial binary relation in a ring.

Commutative semirings
A semiring is called a commutative semiring if also the multiplication is commutative. Its axioms can be stated concisely: It consists of two commutative monoids $$\langle +, 0\rangle$$ and $$\langle \cdot, 1\rangle$$ on one set such that $$a\cdot 0 = 0$$ and $$a\cdot (b+c)=a\cdot b + a\cdot c$$.

The center of a semiring is a sub-semiring and being commutative is equivalent to being its own center.

The commutative semiring of natural numbers is the initial object among its kind, meaning there is a unique structure preserving map of $${\mathbb N}$$ into any commutative semiring.

The bounded distributive lattices are partially ordered, commutative semirings fulfilling certain algebraic equations relating to distributivity and idempotence. Thus so are their duals.

Ordered semirings
Notions or order can be defined using strict, non-strict or second-order formulations. Additional properties such as commutativity simplify the axioms.

Given a strict total order (also sometimes called linear order, or pseudo-order in a constructive formulation), then by definition, the positive and negative elements fulfill $$0<x$$ resp. $$x<0$$. By irreflexivity of a strict order, if $$s$$ is a left zero divisor, then $$s\cdot x < s\cdot y$$ is false. The non-negative elements are characterized by $$\neg(x<0)$$, which is then written $$0\le x$$.

Generally, the strict total order can be negated to define an associated partial order. The asymmetry of the former manifests as $$x<y\to x\le y$$. In fact in classical mathematics the latter is a (non-strict) total order and such that $$0\le x$$ implies $$x=0\lor 0<x$$. Likewise, given any (non-strict) total order, its negation is irreflexive and transitive, and those two properties found together are sometimes called strict quasi-order. Classically this defines a strict total order – indeed strict total order and total order can there be defined in terms of one another.

Recall that "$$\le_\text{pre}$$" defined above is trivial in any ring. The existence of rings that admit a non-trivial non-strict order shows that these need not necessarily coincide with "$$\le_\text{pre}$$".

Additively idempotent semirings
A semiring in which every element is an additive idempotent, that is, $$x+x=x$$ for all elements $$x$$, is called an (additively) idempotent semiring. Establishing $$1 + 1 = 1$$ suffices. Be aware that sometimes this is just called idempotent semiring, regardless of rules for multiplication.

In such a semiring, $$x\le_\text{pre} y$$ is equivalent to $$x + y = y$$ and always constitutes a partial order, here now denoted $$x\le y$$. In particular, here $$x \le 0\leftrightarrow x = 0$$. So additively idempotent semirings are zerosumfree and, indeed, the only additively idempotent semiring that has all additive inverses is the trivial ring and so this property is specific to semiring theory. Addition and multiplication respect the ordering in the sense that $$x \le y$$ implies $$x + t \leq y + t$$, and furthermore implies $$s\cdot x \le s\cdot y$$ as well as $$x\cdot s \le y\cdot s$$, for all $$x, y, t$$ and $$s$$.

If $$R$$ is additively idempotent, then so are the polynomials in $$R[X^*]$$.

A semiring such that there is a lattice structure on its underlying set is lattice-ordered if the sum coincides with the meet, $$x + y = x\lor y$$, and the product lies beneath the join $$x\cdot y \le x\land y$$. The lattice-ordered semiring of ideals on a semiring is not necessarily distributive with respect to the lattice structure.

More strictly than just additive idempotence, a semiring is called simple iff $$x+1=1$$ for all $$x$$. Then also $$1+1=1$$ and $$x \le 1$$ for all $$x$$. Here $$1$$ then functions akin to an additively infinite element. If $$R$$ is an additively idempotent semiring, then $$\{x\in R\mid x+1=1\}$$ with the inherited operations is its simple sub-semiring. An example of an additively idempotent semiring that is not simple is the tropical semiring on $${\mathbb R}\cup\{-\infty\}$$ with the 2-ary maximum function, with respect to the standard order, as addition. Its simple sub-semiring is trivial.

A c-semiring is an idempotent semiring and with addition defined over arbitrary sets.

An additively idempotent semiring with idempotent multiplication, $$x^2=x$$, is called additively and multiplicatively idempotent semiring, but sometimes also just idempotent semiring. The commutative, simple semirings with that property are exactly the bounded distributive lattices with unique minimal and maximal element (which then are the units). Heyting algebras are such semirings and the Boolean algebras are a special case.

Further, given two bounded distributive lattices, there are constructions resulting in commutative additively-idempotent semirings, which are more complicated than just the direct sum of structures.

Number lines
In a model of the ring $${\mathbb R}$$, one can define a non-trivial positivity predicate $$0<x$$ and a predicate $$x<y$$ as $$0<(y-x)$$ that constitutes a strict total order, which fulfills properties such as $$\neg(x<0 \lor 0<x) \to x=0$$, or classically the law of trichotomy. With its standard addition and multiplication, this structure forms the strictly ordered field that is Dedekind-complete. By definition, all first-order properties proven in the theory of the reals are also provable in the decidable theory of the real closed field. For example, here $$ x < y $$ is mutually exclusive with $$\exists d. y + d^2 = x$$.

But beyond just ordered fields, the four properties listed below are also still valid in many sub-semirings of $${\mathbb R}$$, including the rationals, the integers, as well as the non-negative parts of each of these structures. In particular, the non-negative reals, the non-negative rationals and the non-negative integers are such a semirings. The first two properties are analogous to the property valid in the idempotent semirings: Translation and scaling respect these ordered rings, in the sense that addition and multiplication in this ring validate In particular, $$(0<y\land 0<s)\to 0 < s\cdot y$$ and so squaring of elements preserves positivity.
 * $$(x<y)\,\to\,x+t<y+t$$
 * $$(x<y\land 0<s)\,\to\,s\cdot x < s\cdot y$$

Take note of two more properties that are always valid in a ring. Firstly, trivially $$P\,\to\,x \le_\text{pre} y$$ for any $$P$$. In particular, the positive additive difference existence can be expressed as Secondly, in the presence of a trichotomous order, the non-zero elements of the additive group are partitioned into positive and negative elements, with the inversion operation moving between them. With $$(-1)^2=1$$, all squares are proven non-negative. Consequently, non-trivial rings have a positive multiplicative unit, Having discussed a strict order, it follows that $$0\neq 1$$ and $$1\neq 1+1$$, etc.
 * $$(x<y)\,\to\,x \le_\text{pre} y$$
 * $$0<1$$

Discretely ordered semirings
There are a few conflicting notions of discreteness in order theory. Given some strict order on a semiring, one such notion is given by $$1$$ being positive and covering $$0$$, i.e. there being no element $$x$$ between the units, $$\neg(0<x \land x<1)$$. Now in the present context, an order shall be called discrete if this is fulfilled and, furthermore, all elements of the semiring are non-negative, so that the semiring starts out with the units.

Denote by $${\mathsf {PA}}^-$$ the theory of a commutative, discretely ordered semiring also validating the above four properties relating a strict order with the algebraic structure. All of its models have the model $$\N$$ as its initial segment and Gödel incompleteness and Tarski undefinability already apply to $${\mathsf {PA}}^-$$. The non-negative elements of a commutative, discretely ordered ring always validate the axioms of $${\mathsf {PA}}^-$$. So a slightly more exotic model of the theory is given by the positive elements in the polynomial ring $${\mathbb Z}[X]$$, with positivity predicate for $$ p={\textstyle\sum}_{k=0}^n a_k X^k $$ defined in terms of the last non-zero coefficient, $$0 < p := (0 < a_n) $$, and $$p < q := (0 < q - p)$$ as above. While $${\mathsf {PA}}^-$$ proves all $\Sigma_1$-sentences that are true about $$\N$$, beyond this complexity one can find simple such statements that are independent of $${\mathsf {PA}}^-$$. For example, while $$\Pi_1$$-sentences true about $$\N$$ are still true for the other model just defined, inspection of the polynomial $$X$$ demonstrates $${\mathsf {PA}}^-$$-independence of the $$\Pi_2$$-claim that all numbers are of the form $$2q$$ or $$2q+1$$ ("odd or even"). Showing that also $${\mathbb Z}[X,Y]/(X^2-2Y^2)$$ can be discretely ordered demonstrates that the $$\Pi_1$$-claim $$x^2\neq 2y^2$$ for non-zero $$x$$ ("no rational squared equals $$2$$") is independent. Likewise, analysis for $${\mathbb Z}[X,Y,Z]/(XZ-Y^2)$$ demonstrates independence of some statements about factorization true in $$\N$$. There are $${\mathsf {PA}}$$ characterizations of primality that $${\mathsf {PA}}^-$$ does not validate for the number $$2$$.

In the other direction, from any model of $${\mathsf {PA}}^-$$ one may construct an ordered ring, which then has elements that are negative with respect to the order, that is still discrete the sense that $$1$$ covers $$0$$. To this end one defines an equivalence class of pairs from the original semiring. Roughly, the ring corresponds to the differences of elements in the old structure, generalizing the way in which the initial ring $$\Z$$ can be defined from $$\N$$. This, in effect, adds all the inverses and then the preorder is again trivial in that $$\forall x. x\le_\text{pre} 0$$.

Beyond the size of the two-element algebra, no simple semiring starts out with the units. Being discretely ordered also stands in contrast to, e.g., the standard ordering on the semiring of non-negative rationals $${\mathbb Q}_{\ge 0}$$, which is dense between the units. For another example, $${\mathbb Z}[X]/(2X^2-1)$$ can be ordered, but not discretely so.

Natural numbers
$${\mathsf {PA}}^-$$ plus mathematical induction gives a theory equivalent to first-order Peano arithmetic $${\mathsf {PA}}$$. The theory is also famously not categorical, but $$\N$$ is of course the intended model. $${\mathsf {PA}}$$ proves that there are no zero divisors and it is zerosumfree and so no model of it is a ring.

The standard axiomatization of $${\mathsf {PA}}$$ is more concise and the theory of its order is commonly treated in terms of the non-strict "$$\le_\text{pre}$$". However, just removing the potent induction principle from that axiomatization does not leave a workable algebraic theory. Indeed, even Robinson arithmetic $${\mathsf {Q}}$$, which removes induction but adds back the predecessor existence postulate, does not prove the monoid axiom $$\forall y. (0 + y = y)$$.

Complete semirings
A complete semiring is a semiring for which the additive monoid is a complete monoid, meaning that it has an infinitary sum operation $$\Sigma_I$$ for any index set $$I$$ and that the following (infinitary) distributive laws must hold:
 * $${\textstyle\sum}_{i \in I}{\left(a \cdot a_i\right)} = a \cdot \left({\textstyle\sum}_{i \in I}{a_i}\right), \qquad {\textstyle\sum}_{i \in I}{\left(a_i \cdot a\right)} = \left({\textstyle\sum}_{i \in I}{a_i}\right) \cdot a.$$

Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring. For commutative, additively idempotent and simple semirings, this property is related to residuated lattices.

Continuous semirings
A continuous semiring is similarly defined as one for which the addition monoid is a continuous monoid. That is, partially ordered with the least upper bound property, and for which addition and multiplication respect order and suprema. The semiring $$\N \cup \{ \infty \}$$ with usual addition, multiplication and order extended is a continuous semiring.

Any continuous semiring is complete: this may be taken as part of the definition.

Star semirings
A star semiring (sometimes spelled starsemiring) is a semiring with an additional unary operator $${}^*$$, satisfying
 * $$a^* = 1 + a a^* = 1 + a^* a.$$

A Kleene algebra is a star semiring with idempotent addition and some additional axioms. They are important in the theory of formal languages and regular expressions.

Complete star semirings
In a complete star semiring, the star operator behaves more like the usual Kleene star: for a complete semiring we use the infinitary sum operator to give the usual definition of the Kleene star:
 * $$a^* = {\textstyle\sum}_{j \geq 0}{a^j},$$

where
 * $$a^j = \begin{cases}

1,                                & j = 0,\\ a \cdot a^{j-1} = a^{j-1} \cdot a, & j > 0. \end{cases}$$ Note that star semirings are not related to *-algebra, where the star operation should instead be thought of as complex conjugation.

Conway semiring
A Conway semiring is a star semiring satisfying the sum-star and product-star equations:
 * $$\begin{align}

(a + b)^* &= \left(a^* b\right)^* a^*, \\ (ab)^* &= 1 + a(ba)^* b. \end{align}$$

Every complete star semiring is also a Conway semiring, but the converse does not hold. An example of Conway semiring that is not complete is the set of extended non-negative rational numbers $$\Q_{\geq 0} \cup \{ \infty \}$$ with the usual addition and multiplication (this is a modification of the example with extended non-negative reals given in this section by eliminating irrational numbers). An iteration semiring is a Conway semiring satisfying the Conway group axioms, associated by John Conway to groups in star-semirings.

Examples

 * By definition, any ring and any semifield is also a semiring.
 * The non-negative elements of a commutative, discretely ordered ring form a commutative, discretely (in the sense defined above) ordered semiring. This includes the non-negative integers $$\N$$.
 * Also the non-negative rational numbers as well as the non-negative real numbers form commutative, ordered semirings. The latter is called . Neither are rings or distributive lattices. These examples also have multiplicative inverses.
 * New semirings can conditionally be constructed from existing ones, as described. The extended natural numbers $$\N \cup \{ \infty \}$$ with addition and multiplication extended so that $$0 \cdot \infty = 0$$.
 * The set of polynomials with natural number coefficients, denoted $$\N[x],$$ forms a commutative semiring. In fact, this is the free commutative semiring on a single generator $$\{ x \}.$$ Also polynomials with coefficients in other semirings may be defined, as discussed.
 * The non-negative terminating fractions $$\tfrac{\N}{b^{\N}} := \left\{ mb^{-n} \mid m, n \in \N \right\}$$, in a positional number system to a given base $$b\in \N$$, form a sub-semiring of the rationals. One has $$\tfrac{\N}{b^{\N}} \subseteq \tfrac{\N}{c^{\N}}$$ if $$b$$ divides $$c$$. For $$|b| > 1$$, the set $$\tfrac{\Z_0}{b^{\Z_0}} := \tfrac{\N}{b^{\N}} \cup \left(-\tfrac{\N_0}{b^{\N}}\right) $$ is the ring of all terminating fractions to base $$b,$$ and is dense in $$\Q$$.
 * The log semiring on $$\R \cup \{ \pm \infty \}$$ with addition given by $$ x \oplus y = - \log\left(e^{-x} + e^{-y}\right)$$ with multiplication $$+,$$ zero element $$+ \infty,$$ and unit element $$0.$$

Similarly, in the presence of an appropriate order with bottom element,
 * Tropical semirings are variously defined. The semiring $$\R \cup \{ - \infty \}$$ is a commutative semiring with $$\max(a, b)$$ serving as semiring addition (identity $$- \infty$$) and ordinary addition (identity 0) serving as semiring multiplication.  In an alternative formulation, the tropical semiring is $$\R \cup \{ \infty \},$$ and min replaces max as the addition operation.  A related version has $$\R \cup \{ \pm \infty \}$$ as the underlying set. They are an active area of research, linking algebraic varieties with piecewise linear structures.
 * The Łukasiewicz semiring: the closed interval $$[0, 1]$$ with addition of $$a$$ and $$b$$ given by taking the maximum of the arguments ($$\max(a, b)$$) and multiplication of $$a$$ and $$b$$ given by $$\max(0, a + b - 1)$$ appears in multi-valued logic.
 * The Viterbi semiring is also defined over the base set $$[0, 1]$$ and has the maximum as its addition, but its multiplication is the usual multiplication of real numbers. It appears in probabilistic parsing.

Note that $$\max(x, x) = x$$. More regarding additively idempotent semirings,
 * The set of all ideals of a given semiring form a semiring under addition and multiplication of ideals.
 * Any bounded, distributive lattice is a commutative, semiring under join and meet. A Boolean algebra is a special case of these. A Boolean ring is also a semiring (indeed, a ring) but it is not idempotent under . A  is a semiring isomorphic to a sub-semiring of a Boolean algebra.
 * The commutative semiring formed by the two-element Boolean algebra and defined by $$1 + 1 = 1$$. It is also called the . Now given two sets $$X$$ and $$Y,$$ binary relations between $$X$$ and $$Y$$ correspond to matrices indexed by $$X$$ and $$Y$$ with entries in the Boolean semiring, matrix addition corresponds to union of relations, and matrix multiplication corresponds to composition of relations.
 * Any unital quantale is a semiring under join and multiplication.
 * A normal skew lattice in a ring $$R$$ is a semiring for the operations multiplication and nabla, where the latter operation is defined by $$a \nabla b = a + b + ba - aba - bab$$

More using monoids,
 * The construction of semirings $$\operatorname{End}(M)$$ from a commutative monoid $$M$$ has been described. As noted, give a semiring $$R$$, the $$n\times n$$ matrices form another semiring. For example, the matrices with non-negative entries, $$_n(\N),$$ form a matrix semiring.
 * Given an alphabet (finite set) Σ, the set of formal languages over $$\Sigma$$ (subsets of $\Sigma^*$) is a semiring with product induced by string concatenation $$L_1 \cdot L_2 = \left\{ w_1 w_2 \mid w_1 \in L_1, w_2 \in L_2 \right\}$$ and addition as the union of languages (that is, ordinary union as sets). The zero of this semiring is the empty set (empty language) and the semiring's unit is the language containing only the empty string.
 * Generalizing the previous example (by viewing $$\Sigma^*$$ as the free monoid over $$\Sigma$$), take $$M$$ to be any monoid; the power set $$\wp(M)$$ of all subsets of $$M$$ forms a semiring under set-theoretic union as addition and set-wise multiplication: $$U \cdot V = \{ u \cdot v \mid u \in U,\ v \in V \}.$$
 * Similarly, if $$(M, e, \cdot)$$ is a monoid, then the set of finite multisets in $$M$$ forms a semiring. That is, an element is a function $$f \mid M \to \N$$; given an element of $$M,$$ the function tells you how many times that element occurs in the multiset it represents. The additive unit is the constant zero function. The multiplicative unit is the function mapping $$e$$ to $$1,$$ and all other elements of $$M$$ to $$0.$$ The sum is given by $$(f + g)(x) = f(x) + g(x)$$ and the product is given by $$(fg)(x) = \sum\{ f(y) g(z) \mid y \cdot z = x \}.$$

Regarding sets and similar abstractions,
 * Given a set $$U,$$ the set of binary relations over $$U$$ is a semiring with addition the union (of relations as sets) and multiplication the composition of relations. The semiring's zero is the empty relation and its unit is the identity relation. These relations correspond to the matrix semiring (indeed, matrix semialgebra) of square matrices indexed by $$U$$ with entries in the Boolean semiring, and then addition and multiplication are the usual matrix operations, while zero and the unit are the usual zero matrix and identity matrix.
 * The set of cardinal numbers smaller than any given infinite cardinal form a semiring under cardinal addition and multiplication. The class of of an inner model form a (class) semiring under (inner model) cardinal addition and multiplication.
 * The family of (isomorphism equivalence classes of) combinatorial classes (sets of countably many objects with non-negative integer sizes such that there are finitely many objects of each size) with the empty class as the zero object, the class consisting only of the empty set as the unit, disjoint union of classes as addition, and Cartesian product of classes as multiplication.
 * Isomorphism classes of objects in any distributive category, under coproduct and product operations, form a semiring known as a Burnside rig. A Burnside rig is a ring if and only if the category is trivial.

Star semirings
Several structures mentioned above can be equipped with a star operation.
 * The aforementioned semiring of binary relations over some base set $$U$$ in which $$R^* = \bigcup_{n \geq 0} R^n$$ for all $$R\subseteq U \times U.$$ This star operation is actually the reflexive and transitive closure of $$R$$ (that is, the smallest reflexive and transitive binary relation over $$U$$ containing $$R.$$).
 * The semiring of formal languages is also a complete star semiring, with the star operation coinciding with the Kleene star (for sets/languages).
 * The set of non-negative extended reals $$[0, \infty]$$ together with the usual addition and multiplication of reals is a complete star semiring with the star operation given by $$a^* = \tfrac{1}{1 - a}$$ for $$0 \leq a < 1$$ (that is, the geometric series) and $$a^* = \infty$$ for $$a \geq 1.$$
 * The Boolean semiring with $$0^* = 1^* = 1.$$
 * The semiring on $$\N \cup \{ \infty \},$$ with extended addition and multiplication, and $$0^* = 1, a^* = \infty$$ for $$a \geq 1.$$

Applications
The $$(\max, +)$$ and $$(\min, +)$$ tropical semirings on the reals are often used in performance evaluation on discrete event systems. The real numbers then are the "costs" or "arrival time"; the "max" operation corresponds to having to wait for all prerequisites of an events (thus taking the maximal time) while the "min" operation corresponds to being able to choose the best, less costly choice; and + corresponds to accumulation along the same path.

The Floyd–Warshall algorithm for shortest paths can thus be reformulated as a computation over a $$(\min, +)$$ algebra. Similarly, the Viterbi algorithm for finding the most probable state sequence corresponding to an observation sequence in a hidden Markov model can also be formulated as a computation over a $$(\max, \times)$$ algebra on probabilities. These dynamic programming algorithms rely on the distributive property of their associated semirings to compute quantities over a large (possibly exponential) number of terms more efficiently than enumerating each of them.

Generalizations
A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a semigroup rather than a monoid. Such structures are called or. A further generalization are, which additionally do not require right-distributivity (or , which do not require left-distributivity).

Yet a further generalization are : in addition to not requiring a neutral element for product, or right-distributivity (or left-distributivity), they do not require addition to be commutative. Just as cardinal numbers form a (class) semiring, so do ordinal numbers form a near-semiring, when the standard ordinal addition and multiplication are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead.

In category theory, a is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig.

title: Semiring


text:
Tangelo

The tangelo (, ; C. reticulata × C. maxima or × C. paradisi), Citrus × tangelo, is a citrus fruit hybrid of a Citrus reticulata variety, such as mandarin orange or tangerine, and a Citrus maxima variety, such as a pomelo or grapefruit. The name is a portmanteau of 'tangerine' and 'pomelo'.

Tangelos are the size of an adult fist, have a tart and tangy taste, and are juicy at the expense of flesh. They generally have loose skin and are easier to peel than oranges, readily distinguished from them by a characteristic "nipple" at the stem. Tangelos can be used as a substitute for mandarin oranges or sweet oranges.

Orlando
The early maturing Orlando tangelo is noted for its rich juiciness, mild and sweet flavor, large size, distinct zesty smell, and flat-round shape without a characteristic knob. California/Arizona tangelos have a slightly pebbled texture, vibrant interior and exterior color, very few seeds, and a tight-fitting rind. Orlando tangelos are available from mid-November to the beginning of February. The tangelo originated as a cross between a Duncan grapefruit and a Dancy tangerine. Walter Tennyson Swingle of the United States Department of Agriculture (USDA) is credited with creating the hybrid in 1911. When the Orlando tangelo was first cultivated, it was known by the name Lake tangelo. The trees of this variety grow to a large size and are easily recognized by their cup-shaped leaves. Orlando tangelos are recognized as one of the more cold-tolerant varieties. Northern Florida grows significantly fewer tangelos, but they are much sweeter due to climate.

Minneola
The Minneola tangelo (also known as the Honeybell) is a cross between a Duncan grapefruit and a Dancy tangerine and was released in 1931 by the USDA Horticultural Research Station in Orlando. It is named after Minneola, Florida. Most Minneola tangelos are characterized by a stem-end neck, which tends to make the fruit appear bell-shaped. Because of this, it is also called the Honeybell in the gift fruit trade, one of the most popular varieties. Honeybell is sometimes used as unofficial shorthand for premium cultivation. Minneolas are usually fairly large, typically 3–3+1/2 in in diameter. The peel color, when mature, is a bright-reddish-orange color. The rind of the Minneola is relatively thin. Minneolas peel rather easily and are very juicy. The Minneola is not strongly self-fruitful, and yields will be greater when interplanted with suitable pollenizers such as Temple tangor, Sunburst tangerine, or possibly Fallglo tangerine. It tends to bear a good crop every other year. In the Northern Hemisphere the fruit matures in the December–February period, with January being the peak.

Jamaican tangelo
The Jamaican tangelo, marketed under proprietary names 'ugli fruit' and 'uniq fruit,' is a spontaneous hybrid discovered about 1920 on the island of Jamaica, with a rough, wrinkled, greenish-yellow rind. Its exact parentage has not been determined, but it is thought to be a tangerine/grapefruit hybrid.

K-Early (Sunrise)
A hybrid propagated by Walter Tennyson Swingle and Herbert John Webber, the K-Early is an early-ripening cultivar that gained a bad reputation at first but has been increasing in popularity in recent years. It is sometimes called 'Sunrise,' a name also used for a different and older cultivar.

Mapo
The Mapo (a portmanteau between 'Mandarino' and the italian word for grapefruit, 'Pompelmo') is a hybrid developed in 1950 in Italy by the Italian Ministry of Agriculture Citrus Research Station of Acireale. In Italy, the Mapo matures at the end of summer, about two months earlier than most citruses. Its peel is green, smooth, and thin; yellow markings appear when fully ripened. Its pulp is yellow, with orange nuances also when fully ripened. It is a cross between the 'Avana' mandarin and the Duncan grapefruit.

Seminole
The seminole is a hybrid between a 'Bowen' grapefruit and a 'Dancy' tangerine. It is deep red-orange in color, oblate in shape with a thin and firm peel, and is not necked. It has 11-13 juicy segments and a pleasant, subacid flavor. It has 20-25 small seeds. The tree is high-yielding and scab-resistant.

Thornton
A tangerine-grapefruit hybrid developed by Walter Tennyson Swingle in 1899, the Thornton is oblate to obovate, slightly rough, and medium to large in size. The peel is light orange and is of medium thickness; the pulp inside is pale to deep orange. It has 10-12 juicy segments and a rich subacid to sweet flavor. There are 10-25 slender seeds inside. It ripens from December to March. The tree is high-yielding and is well-adapted to hot and dry regions, although the fruit ships poorly.

Novel varieties
In 2011, a troop of baboons was attracted to the higher sweetness of a new likely mutation in a Minneola planting in Cape Town, South Africa, prompting its propagation.

Drug interactions
One study thus far has shown that, unlike grapefruit, interactions with statins are not likely with tangelos. Although the tangelo is derived from a grapefruit crossed with a mandarin, the furocoumarins in grapefruit are not expressed in tangelos.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Tangelo


text:
Ruud van Nistelrooy

Rutgerus Johannes Martinus van Nistelrooij (born 1 July 1976), commonly known as Ruud van Nistelrooy, is a Dutch professional football coach and former striker who is currently an assistant manager at Manchester United. Widely regarded as one of the best strikers of all time, Van Nistelrooy has been the top scorer in three separate UEFA Champions League seasons, and is the sixth-highest goalscorer in the competition's history with 56 goals. He has also been the top scorer in three different European domestic leagues. In 2004, he was listed in the FIFA 100 of the world's greatest living players.

Van Nistelrooy began his career with Den Bosch, before moving onto Heerenveen, eventually making a name for himself at PSV Eindhoven, where he won two Eredivisie titles. His goalscoring record at PSV attracted attention from Manchester United; a deal was in place in the summer of 2000, but because of injury problems, his move was secured a year later for a then British record fee of £19 million. His time at United was successful, winning the Premier League, FA Cup, Football League Cup, and FA Community Shield, along with winning the Sir Matt Busby Player of the Year twice. Van Nistelrooy scored 150 goals in just 219 games for United, as well as becoming at the time their all-time European record goalscorer, but fell out of favour towards the end of his tenure. Real Madrid then secured his services in 2006. Although an injury blighted the end of his days with Madrid, he won La Liga twice and the Supercopa de España before signing for Hamburger SV during the January transfer window in 2010. After one-and-a-half seasons with Hamburg, he moved back to Spain with Málaga in the summer of 2011. His retirement from football was announced on 14 May 2012.

Internationally, Van Nistelrooy represented the Netherlands on 70 occasions, netting 35 times. He was part of the Euro 2004, 2006 World Cup, and Euro 2008 squads.

Early career
Born in Oss, North Brabant, Van Nistelrooy started his career in 1993, aged 17, with Dutch Eerste Divisie side Den Bosch, where he was converted from a central midfielder to centre-forward, after playing for local clubs Nooit Gedacht and RKSV Margriet. After netting 12 goals in 31 games in the 1996–97 campaign, he transferred for €360,000 to Heerenveen the next year, and scored 13 goals in 31 matches in his only season with the club. He was then signed by PSV Eindhoven the next season for €6.3 million, a then-record transfer sum between two Dutch teams.

He scored 31 goals in 34 matches, the highest season total in the Eredivisie and second-highest in Europe overall, in addition to scoring all three of PSV's goals in a Champions League match against HJK Helsinki on 25 November 1998. Van Nistelrooy capped off the year by winning the Dutch Player of the Year award. The following season, he won his second Eredivisie scoring title with 29 goals. According to a 2001 interview with The Daily Telegraph, Manchester United coach Sir Alex Ferguson said that his son, Darren, who was at try-outs for Eredivisie rival Heerenveen at the time, begged his father, "You've got to sign Van Nistelrooy right away, he's fantastic. We've been watching him." Ferguson sent team representatives to PSV's next league game and signed Van Nistelrooy the next day.

Manchester United
Van Nistelrooy looked set to complete a club record £18.5 million transfer to Manchester United in the summer of 2000. He was to be unveiled at a press conference four days later, but this was instead used to announce that the transfer had been postponed over concerns about his fitness, having not played for a month due to problems with his knee. The transfer was then cancelled after PSV refused to agree to further medical tests, and the next day he suffered a rupture to his anterior cruciate knee ligaments during a training session, leaving him injured for a year.

A year later, Van Nistelrooy signed a five-year contract after passing his medical. He downplayed United's £19 million investment to reporters, saying, "The price is not heavy for me – it lifts me up because it means United have big confidence in me."

2001–02
Van Nistelrooy made his debut in the Charity Shield against Liverpool, scoring in a 2–1 loss. On 19 August, Van Nistelrooy made his Premier League debut against Fulham at Old Trafford, and scored two goals to help United to a 3–2 win. He scored his first Champions League goals on 17 October in a 3–2 loss to Deportivo La Coruña.

On 22 December, Van Nistelrooy scored his first Premier League hat-trick in a 6–1 win against Southampton. In the FA Cup, Van Nistelrooy started on the bench for the third round tie against Aston Villa due to a groin strain, but was substituted on for Luke Chadwick in the second-half with United 2–0 down and scored two "sublime" goals in three minutes to help United advance to the fourth round with a 3–2 win.

In total during his first season, Van Nistelrooy scored 23 goals in 32 league games. He broke the record he shared with Mark Stein, Alan Shearer and Thierry Henry, by scoring in eight consecutive league games. He also scored ten Champions League goals, and was named the PFA Players' Player of the Year.

2002–03
The following season, he finished as the top Premier League scorer with 25 goals in 34 games, with a goal on the final day against Everton earning him the Golden Boot ahead of Arsenal's Thierry Henry. His tally included three hat-tricks against Newcastle United, Fulham, and Charlton Athletic. He also scored in each of the final eight games of the season as United won the title. Van Nistelrooy won the Premier League Player of the Season in 2003. He was accredited with spearheading the title win for United with a goal tally that was described as "astonishing". After scoring 12 Champions League goals in nine consecutive matches, he was named by UEFA as the best striker in Europe.

2003–04

Van Nistelrooy started the 2003–04 season by scoring twice in his first two league matches. This meant he had scored in 10 consecutive league games, then a Premier League record, which stood until 2015 when Jamie Vardy scored in 11 consecutive matches. In the 0–0 draw on 21 September 2003, later dubbed the Battle of Old Trafford, Van Nistelrooy was at the centre of one of the most infamous incidents of the Arsenal–Manchester United rivalry. After missing a last-minute penalty that would have won the game for United, Van Nistelrooy was attacked by Martin Keown, prompting a melee involving several players from both sides; five Arsenal players including Keown received various bans and fines, while the club itself received a record fine of £175,000. Earlier in the game, he had been fouled by Patrick Vieira, who was sent off for a second bookable offence. Vieira and his manager Arsène Wenger both accused Van Nistelrooy of cheating and stamping on Vieira, prompting a vociferous defence by Alex Ferguson.

On 27 September, he scored a hat-trick against Leicester City in a 4–1 victory. He scored his 100th goal for the club in a dramatic 4–3 victory over Everton on 7 February 2004.

In the FA Cup fifth round derby against Manchester City, Van Nistelrooy scored twice to help ten-man United win 4–2, and again scored twice in the quarter-final against Fulham as United came from behind to win 2–1. He scored a further two goals, including one penalty, in United's victory over Millwall in the 2004 FA Cup Final.

2004–05
Van Nistelrooy missed a large part of the 2004–05 season due to injury, but nonetheless scored a Champions League-best eight goals. One of them was his 30th career European goal, which he scored in a 2–2 Champions League group stage draw with Lyon on 15 September 2004, overtaking Denis Law's previous club record of 28 goals. Law later said to reporters, "I'm delighted for Ruud. It could not happen to a nicer guy."

On 24 October, Arsenal returned to Old Trafford for another dramatic fixture. Arsenal entered the fixture with 49-game unbeaten run and had been dubbed "The Invincibles", but United won the game 2–0. Van Nistelrooy scored the opening goal from the penalty spot, atoning for the penalty miss the previous year, with Wayne Rooney scoring a second. Van Nistelrooy was later banned for three games for a foul on Ashley Cole that the referee had missed. On 3 November, Van Nistelrooy scored all four goals in a 4–1 win against Sparta Prague. On 17 April 2005, he scored two goals in the FA Cup semi-final in a 4–1 win against Newcastle, but United lost the final on penalties to Arsenal.

2005–06

At the start of the 2005–06 season, Van Nistelrooy scored in United's first four Premier League games. He finished as the second-highest league scorer with 21 goals, behind Arsenal's Thierry Henry. However, he was benched for the League Cup Final against Wigan Athletic, fuelling speculation of a rift between him and Alex Ferguson, which Van Nistelrooy denied. He was nonetheless left on the bench for six consecutive league matches, and though he then returned to the starting line-up and scored match-winners against West Ham United and Bolton Wanderers, fresh doubt spread over Van Nistelrooy's future when he was benched for United's season finale win over Charlton Athletic. Ferguson claimed that Van Nistelrooy was angry at the decision and left the stadium three hours before kick-off. On 9 May 2006, it was reported that Van Nistelrooy's exclusion from the squad was due to a training session fight between him and teammate Cristiano Ronaldo. Van Nistelrooy allegedly criticised Ronaldo's tendency to hold onto the ball instead of passing to his teammates, which sparked the fight, after which Van Nistelrooy remarked, "What are you going to do? Complain to your daddy?" This was in reference to United's Portuguese assistant manager, Carlos Queiroz, but Ronaldo took the statement literally and allegedly broke into tears since his father, José had died eight months previously. Van Nistelrooy later apologised to Ferguson for his behaviour in the previous few months.

Van Nistelrooy signed with Spanish La Liga side Real Madrid on 28 July, departing Old Trafford after five seasons with a total of 150 goals in 219 appearances, with his final goal coming on 1 April against Bolton, scoring what proved to be the winning goal. With 38 goals, he is also the club's second all-time European goalscorer, behind Wayne Rooney.

Real Madrid

On 15 July 2006, Ferguson confirmed that Van Nistelrooy wanted to leave Manchester United and Real Madrid announced two weeks later that he had signed a three-year contract after being purchased for €14 million.

Van Nistelrooy made his debut for the Spanish club in a 1–0 friendly win over Reggina on 4 August 2006. Van Nistelrooy scored a hat-trick in his second league match against Levante and, on 12 November 2006, he scored all four of Real Madrid's goals in a 4–1 victory over Osasuna. For the first six months he was a teammate of Ronaldo, a player whom Van Nistelrooy regards as the best he ever played with. Van Nistelrooy won the league's Pichichi award with 25 goals as Real Madrid took home the 2006–07 title, and he also equalled the longest consecutive scoring streak in La Liga history with seven straight matches, tying a league record shared by Hugo Sánchez.

In January 2008, Van Nistelrooy signed a contract extension keeping him with Madrid until 2010, with the expiration date one day shy of his 34th birthday. He underwent ankle surgery in March, and returned for the El Clásico derby against Barcelona on 7 May, in which he netted a penalty two minutes after coming on as a substitute. He finished the season with 20 goals in 33 appearances.

In November 2008, Real Madrid announced that Van Nistelrooy would miss the remainder of the 2008–09 season after exploratory arthroscopic surgery revealed a partially torn meniscus in his right knee, with an expected recovery time of six to nine months following a second operation to repair the damage. Van Nistelrooy travelled to the United States to see specialist Richard Steadman, who had previously operated on the same knee back in 2000. At the time of his injury, he had 10 goals in 12 club appearances for the season. Following the injury, he was de-registered by Real Madrid for the rest of the 2008–09 season, and his shirt number was given to Dani Parejo. On 24 August 2009, in the last pre-season match before the start of La Liga, Van Nistelrooy stepped on the pitch for the first time since his injury and played the last 15 minutes of the game against Rosenborg, substituting Kaká. Van Nistelrooy came on for Cristiano Ronaldo in the 80th minute against Xerez in his first La Liga match since recovering from his injury. In the 81st minute, he provided the assist for a Karim Benzema goal, following it with his own 88th-minute goal. During his strike, however, he picked up a thigh injury, where it was later revealed by Real Madrid that he would be out from first team action for up to six weeks. On 27 October, Van Nistelrooy made his second comeback of the season coming on as a substitute for Raúl in the 71st minute against Alcorcón in the Copa del Rey.

Hamburger SV
On 23 January 2010, Van Nistelrooy signed an 18-month contract with the German club Hamburger SV until June 2011. He made his first appearance for the club after coming off the bench in the last two minutes of Hamburg's 3–3 draw with 1. FC Köln on 6 February. Van Nistelrooy scored his first two goals for Hamburg on 13 February 2010 against VfB Stuttgart in the 75th and 77th minutes of a 3–1 win, after entering the game just a few minutes prior. On 11 March 2010, he scored his first UEFA Europa League goal in the 40th minute of Hamburg's match against Anderlecht.

Van Nistelrooy scored his only competitive hat-trick with Hamburg on 15 August 2010, in a 5–1 win over Torgelower SV Greif in the first round of the 2010–11 DFB-Pokal. On 21 August 2010, he scored a brace in Hamburg's season opener against Schalke 04, which HSV won 2–1. During that match, he played against his friend and former Real Madrid teammate Raúl, who was making his Bundesliga debut. In the January transfer window, Van Nistelrooy was linked a return move back to Real Madrid and could be used for a short-term basis to cover, following injuries of Gonzalo Higuaín and Karim Benzema as manager José Mourinho would be happy for Van Nistelrooy to join the club as Mourinho left the door for him. Van Nistelrooy admitted he seriously considered moving back to the club if the bid happened. The club, however, rejected Madrid's bid for Van Nistelrooy, giving the club a warning as Hamburg wants to keep hold of him until the transfer window ended. Despite being angered over his move to Real Madrid rejected, Van Nistelrooy said he would remain committed at Hamburg. During a match between Hamburg and Hannover 96 on 16 April 2011, Van Nistelrooy suffered a calf injury that kept him out until the last matchday.

Van Nistelrooy netted seven goals and made two assists in 25 appearances in his only full season with Hamburg.

Málaga

On 1 June 2011, Van Nistelrooy returned to Spain to pen a one-year deal with La Liga side Málaga, on a free transfer. Van Nistelrooy was presented at La Rosaleda Stadium as he was welcomed by 15,000 Málaga fans. He made his debut in a 2–1 loss against Sevilla on the opening game of the 2011–12 season.

Van Nistelrooy registered his first goal on 1 October 2011 against Getafe. On 21 December, against the same club, he opened the scoring with a volley as Málaga won 3–2 on aggregate in the Copa del Rey. Later on in the season, Van Nistelrooy scored two more goals in the league against Espanyol and Racing Santander, with the latter being his final goal of his career. Just one day before his retirement, Van Nistelrooy made his last appearance of his career, coming on as a substitute in the 75th minute for goalscorer Salomón Rondón, who scored the only goal in Málaga's league match against Sporting Gijón.

On 14 May 2012, Van Nistelrooy announced his retirement from football at the age of 35. He told Sport1 he previously made hints that his football career was coming to an end after insisting Málaga would be his last club.

International career
Van Nistelrooy earned 70 caps and scored 35 goals for the Netherlands. He made his debut for the national team in a friendly match against Germany on 18 November 1998. However, the cruciate ligament injury which postponed his transfer to Manchester United also ruled Van Nistelrooy out of Euro 2000.

As the Dutch failed to qualify for the 2002 World Cup, Van Nistelrooy did not make his tournament debut for the Netherlands until Euro 2004, where he and the Czech Republic's Milan Baroš were the only players to score in all three group stage matches. The Netherlands reached the semi-finals where they were defeated by hosts Portugal 2–1. Van Nistelrooy was named in the tournament all-star team.

During a 2006 World Cup qualifying match, he scored in the Dutch's 4–0 win over bottom-placed Andorra and was yellow-carded for going to Andorra player Antoni Lima and celebrating in front of him, following an incident six minutes before when Van Nistelrooy missed a penalty and Lima laughed at him. He was a part of coach Marco van Basten's squad for the 2006 World Cup finals, for which he served as the official FIFA/SOS ambassador. He started, and was substituted, in all of the Netherlands' group stage matches, and scored his lone goal against the Ivory Coast. Van Nistelrooy was dropped to the bench by Van Basten without an explanation for the Netherlands' second round match, which saw them again eliminated by Portugal.

Van Basten left the striker out of the squad for a friendly against the Republic of Ireland on 16 August 2006. Dirk Kuyt replaced Van Nistelrooy in their next match against Portugal in September. After Klaas-Jan Huntelaar was unavailable for Euro 2008 qualifiers against Bulgaria and Belarus due to injury, Van Nistelrooy refused Van Basten's request to take his place on the roster.

On 23 January 2007, Van Nistelrooy announced his retirement from international football following continued disputes with Van Basten, which had started back in 2006 during the course of the World Cup finals matches. After several phone conversations and at the persuasion of veteran goalkeeper Edwin van der Sar, however, both player and coach put aside their differences. Van Basten announced four months later that Van Nistelrooy was returning to the Oranje. On 8 September, Van Nistelrooy filled Huntelaar's spot, as previously requested, for the qualifier against Bulgaria, scoring in a 2–0 victory, and scored the winning goal four days later in injury time of the Netherlands' win over Albania.

At Euro 2008, Van Nistelrooy scored for the Netherlands in their decisive 3–0 victory over Italy in the group stage, and scored the equaliser in their eventual 3–1 loss to Russia in the quarter-finals. On 4 August, he again announced his retirement from international competition.

During the buildup to the 2010 World Cup, Van Nistelrooy again made himself available, but was left out by the Netherlands' new coach Bert van Marwijk. After being left out for the World Cup, Van Nistelrooy said he had no choice but to face the fact that his international career is finished.

After main striker Robin van Persie suffered an injury during a match with his club Arsenal, Van Marwijk gave Van Nistelrooy a chance to reclaim his position as the main striker of the Netherlands' national team. Van Nistelrooy was invited to two Netherlands' Group E qualification matches for Euro 2012 against San Marino and Finland. He scored in the 5–0 victory against San Marino on 3 September 2010.

Van Nistelrooy was again recalled to the squad in March 2011 for two Euro 2012 qualification games with Hungary, after injuries to strikers Klaas-Jan Huntelaar, Arjen Robben and Theo Janssen. In the away game on 25 March, he appeared as a late substitute for Dirk Kuyt while in the return game in Amsterdam four days later, he came off the bench to score his 35th international goal. adding it is "wonderful to be able to add something to this Oranje team", as he told Berend Scholten.

Style of play
A highly prolific goalscorer, Van Nistelrooy was known for being extremely clinical and opportunistic in front of goal, and stood out throughout his career for his outstanding striking ability and finishing ability with either foot as well as his head, which earned him a reputation as one of the best strikers of his generation. Due to his excellent positional sense, quick reactions, and intelligent offensive movement, he excelled at finding space in the penalty area and at anticipating his opponents to the ball, and was also known for his ability to time his runs to lose his markers, beat the offside trap, and get on the end of his teammates' passes; his tendency to be in the right place at the right time in the box saw him occasionally labelled as a "poacher" in the media.

A well-rounded centre-forward, in addition to his eye for goal, Van Nistelrooy was gifted with pace, physical strength, and shooting power, as well as excellent technical skills, and an ability to hold up the ball with his back to goal and link-up with his teammates or provide them with assists, due to his ability to read the game. He was also an accurate penalty taker. In spite of his talent and goalscoring ability, however, at times he drew criticism from managers, players and pundits for his controversial behaviour, as well as for having a tendency to be selfish, overly flamboyant, or for his penchant for simulation.

PSV under-17s
On 22 June 2013, Van Nistelrooy joined PSV as a coaching intern working with the under-17s. He was revealed as the new striker coach of the under-17s, 19s and the reserve team in February 2016.

Netherlands
In March 2014, it was announced that Van Nistelrooy would be assisting Guus Hiddink after the 2014 World Cup.

PSV under-19s
On 25 June 2018, Van Nistelrooy became the new manager of the PSV under-19s, taking over Mark van Bommel, who left his role for the senior team.

In December 2019 it was confirmed, that Van Nistelrooy beside his position at PSV, also had been added to Ronald Koeman's backroom staff as an assistant manager together with Maarten Stekelenburg ahead of the Euro 2020, after Kees van Wonderen recently had left the position.

PSV first team
In March 2022, Van Nistelrooy was appointed as the replacement for the outgoing Roger Schmidt as PSV's head coach. He signed a three-year contract beginning with the 2022–23 season. After winning the 2023 Johan Cruyff Shield and 2022–23 KNVB Cup, he resigned with one match remaining in the 2022–23 Eredivisie season, citing a lack of support.

Manchester United
On 11 July 2024, Van Nistelrooy was appointed as the new assistant manager for premier League club Manchester United on a two-year contract.

Personal life
Van Nistelrooy married his girlfriend, Leontien Slaats, in July 2004. The couple have a daughter and a son. Van Nistelrooy is a Roman Catholic Christian.

Charity
Van Nistelrooy and his wife are both heavily involved with the charity organization SOS Children's Villages. The organization has been around since 1949 and is an international development charity which serves to protect the interests and rights of children. Van Nistelrooy was officially appointed "FIFA for SOS Children's Villages" Ambassador in the Netherlands on 1 September 2001.

On 17 November 2009, Van Nistelrooy and his wife hosted SOS at Ciudad Real Madrid. The purpose of the event was to help create a calendar which would be sold to benefit the organization.

Endorsements
Van Nistelrooy was sponsored by sportswear company Nike and appeared in Nike commercials. In a global Nike advertising campaign in the run-up to the 2002 World Cup in Korea and Japan, he starred in a "Secret Tournament" commercial (branded "Scorpion KO") directed by Terry Gilliam, appearing alongside football players such as Thierry Henry, Ronaldo, Edgar Davids, Fabio Cannavaro, Francesco Totti, Ronaldinho, Luís Figo and Hidetoshi Nakata, with former player Eric Cantona as the tournament "referee".

Media career
During the 2014 World Cup, Van Nistelrooy worked as a studio analyst for ESPN.

International

 * Scores and results list the Netherlands' goal tally first, score column indicates score after each van Nistelrooy goal.

Player
PSV
 * Eredivisie: 1999–2000, 2000–01
 * Johan Cruyff Shield: 1998

Manchester United
 * Premier League: 2002–03
 * FA Cup: 2003–04
 * Football League Cup: 2005–06
 * FA Community Shield: 2003

Real Madrid
 * La Liga: 2006–07, 2007–08
 * Supercopa de España: 2008

Individual

 * Dutch Footballer of the Year: 1998–99, 1999–2000
 * Eredivisie top goalscorer: 1998–99, 1999–2000
 * Premier League Player of the Month: December 2001, February 2002, April 2003
 * Sir Matt Busby Player of the Year: 2001–02, 2002–03
 * ESM Team of the Year: 2001–02
 * UEFA Champions League top goalscorer: 2001–02, 2002–03, 2004–05
 * UEFA Champions League top assist provider: 2001–02
 * FA Cup top goalscorer: 2002–03, 2003–04
 * PFA Fans' Player of the Year: 2001–02
 * PFA Players' Player of the Year: 2001–02
 * PFA Team of the Year: 2001–02 Premier League, 2003–04 Premier League
 * IFFHS World's Top Goal Scorer of the Year: 2002
 * BBC Goal of the Month: March 2003
 * Premier League Player of the Season: 2002–03
 * Premier League Golden Boot: 2002–03
 * UEFA Club Forward of the Year: 2002–03
 * UEFA Team of the Year: 2003
 * FIFA 100
 * UEFA Euro Team of the Tournament: 2004
 * Pichichi Trophy: 2006–07
 * ESPN World Team of the Decade: 2009
 * IFFHS World's Top Goal Scorer of the Last Decade: 2012

Manager
PSV
 * KNVB Cup: 2022–23
 * Johan Cruyff Shield: 2022

title: Ruud van Nistelrooy


text:
John Canfield Spencer

John Canfield Spencer (January 8, 1788 – May 17, 1855) was an American lawyer, politician, judge and United States Cabinet secretary in the administration of President John Tyler.

After graduating from Union College in 1806, Spencer practiced law and held various positions, including master of chancery, postmaster, and attorney general. Spencer served in the U.S. House of Representatives from 1817 to 1819 and the New York State Assembly and Senate in various years between 1820 and 1833. As an anti-Mason, he investigated the disappearance of William Morgan, which sparked the Anti-Masonic movement.

In 1841, President John Tyler appointed Spencer as Secretary of War, and in 1843, he became Secretary of the Treasury. Spencer faced challenges in his role as Treasury Secretary, including a deficit, tariffs, and the development of a plan for a Board of Exchequer. President Tyler nominated Spencer for open Associate Justice seats on the Supreme Court twice in 1844, but both attempts failed. Spencer resigned as Treasury Secretary in May 1844 and returned to Albany.

Spencer married Elizabeth Scott Smith in 1809, and they had several children, many of whom died young or under unfortunate circumstances.

Early life
John Canfield Spencer was born on January 8, 1788, in Hudson, New York. He was the oldest child of Ambrose Spencer, Chief Justice of the New York Supreme Court, and his first wife, Laura Canfield (1768–1807). His sister, Abby Spencer (1790–1839), was married to Albany Mayor John Townsend. His younger brother, William Augustus Spencer (1792–1854), was married to Eleanora Eliza Lorillard (1801–1843), the daughter of Peter Abraham Lorillard. His brother, Ambrose Spencer, Jr., was killed at the Battle of Lundy's Lane.

After the death of his mother in 1807, his father married Mary Clinton (1773–1808) in 1808. Mary was the daughter of James Clinton and sister of New York Governor DeWitt Clinton. After Mary's death later that same year, his father remarried again to Katherine Clinton (1778–1837), Mary's sister.

He graduated from Union College in 1806, became secretary to New York Governor Daniel D. Tompkins in 1807, studied law in Albany, New York, and was admitted to the bar in 1809.

Career
After commencing practice in Canandaigua, New York, in 1809, Spencer became a master of chancery in 1811.

During the War of 1812, Spencer served in the United States Army where he was appointed brigade judge advocate general for the northern frontier. He was postmaster of Canandaigua, New York in 1814, became assistant attorney general and district attorney for the five western counties of New York in 1815 and was elected a Democratic-Republican to the United States House of Representatives in 1816, serving from March 4, 1817, to March 3, 1819. He was a member of the committee that reported unfavorably on the affairs of the Second Bank of the United States.

In 1819, he was the Clintonian candidate for U.S. Senator from New York, but due to a three-cornered contest with Bucktail Samuel Young and Federalist Rufus King, no-one was elected. He was a member of the New York State Assembly from 1820 to 1822, and was Speaker in 1820. He was a member of the New York State Senate from 1825 to 1828.

In 1826, Spencer served as a special prosecutor to investigate the disappearance of William Morgan who was arrested, kidnapped and murdered for exposing secrets kept by Freemasons, thus sparking the Anti-Masonic movement. Spencer sided with the anti-Masons and was the author of a manuscript on Masonic rituals. He was again a member of the New York Assembly from 1831 to 1833 and moved to Albany, New York in 1837. He edited the English edition of Alexis de Tocqueville's Democracy in America and served as Secretary of State of New York from 1839 to 1841.

Federal government
In 1841, President John Tyler appointed Spencer to be Secretary of War in his administration. As War Secretary, he proposed a chain of posts extending from Council Bluffs, Iowa to the Columbia River. He also recommended that the government adhere to arrangements made by Army commanders in the field for compensation of the Creek Indians, who had been forced to move west of the Mississippi. In 1842, his nineteen-year-old son, Philip Spencer, a midshipman, was executed without court-martial along with two other sailors aboard the brig USS Somers for allegedly attempting mutiny.

In 1843, Spencer was appointed Secretary of the Treasury after the resignation of Walter Forward. As Treasury Secretary, he was preoccupied with the tariff and believed that the deficit and other federal expenditures should be funded by duties on imports rather than by internal taxation, something he was forced to announce for the fiscal year in 1843. The expenditures of the treasury had exceeded its receipts and he advocated additional import duties on articles such as coffee and tea. To help fund the federal deficit he engaged in controversial issues of Treasury Notes. He also continued to develop a plan, originally initiated by Forward, for a Board of Exchequer to keep and disburse public funds raised by duties. The Exchequer bill, which reflected continuing interest in some form of independent treasury system, failed due to a political conflict in the United States Congress.

On two occasions in 1844, President Tyler nominated Spencer to fill open Associate Justice seats on the Supreme Court. The first failed attempt was in January, when Tyler put forward Spencer as a replacement for the recently deceased Smith Thompson. Tyler made the nomination on January 9; on January 31, the Senate rejected Spencer by a 26–21 vote, mainly due to Whig opposition to the president. Tyler then nominated Spencer to fill Henry Baldwin's seat in June but withdrew his name for that of Reuben Hyde Walworth. As one of few northerners in an administration dominated by southern interests, Spencer had found it increasingly difficult to serve in his cabinet post and resigned as Treasury Secretary in May 1844. Thereafter, he returned to Albany.

Personal life
In 1809, Spencer married Elizabeth Scott Smith (1789–1868). Together, they were the parents of several children, many of whom died in infancy or under unfortunate circumstances:

 * Mary Natalie Spencer (1810–1886), who married Henry Morris (1806–1854), son of Capt. Richard Valentine Morris and grandson of Lewis Morris, in 1831.
 * Laura Catherine Spencer (1812–1891), who married George W. Clinton (1807–1885), the 12th Mayor of Buffalo who was the son of DeWitt Clinton.
 * Eliza Abby Spencer (1815–1816), who died young.
 * Ambrose Canfield Spencer (1817–1876), who was murdered in Linn, Missouri, in 1876.
 * John Canfield Spencer, Jr. (d. 1845)
 * Philip Spencer (1823–1842), who was executed for mutiny in 1842.
 * DeWitt Clinton Spencer (1830–1836), who also died young.

In Canandaigua, he lived for 36 years in a house at 210 Main Street, that was built by General Peter Buell Porter (1773–1844), the United States Secretary of War under John Quincy Adams, in about 1800.

He died in Albany, New York, on May 17, 1855. He was interred in Albany Rural Cemetery beside his wife, Elizabeth.

Descendants
His grandson, through his daughter Mary, was Henry Lewis Morris (b. 1845), who married Anna Rutherfurd Russell, the daughter of Archibald Russell and Helen Rutherfurd (née Watts) Russell. They were the parents of Lewis Spencer Morris (b. 1884).

His grandchildren, through his daughter Laura, included Elizabeth Spencer Clinton (1835–1918), Spencer Clinton (1839–1914), Catharine Clinton (1841–1881), and George Clinton (1846–1934).

title: John Canfield Spencer


text:
Stewart Stevenson

James Alexander Stewart Stevenson (Gaelic: Seamus Alasdair Stiùbhart MacSteafain; born 15 October 1946) is a Scottish former politician who served as Minister for Transport, Infrastructure and Climate Change from 2007 to 2010 and Minister for Environment and Climate Change from 2011 to 2012. A member of the Scottish National Party (SNP), he was Member of the Scottish Parliament (MSP) for Banffshire and Buchan Coast, formerly Banff and Buchan, from 2001 to 2021.

Early life
Stevenson was born in Edinburgh. His father James Stevenson was a doctor and his mother Helen MacGregor was a teacher. He was brought up in Cupar, Fife. He was educated at Bell Baxter High School then studied mathematics at the University of Aberdeen. He worked in information technology with the Bank of Scotland for 30 years, retiring in 1999 as Director of Technology Innovation.

In Opposition
Stevenson joined the Scottish National Party (SNP) in 1961. He stood as an SNP candidate in the Linlithgow constituency in the 1999 Scottish General election but was unsuccessful.

In January 2001 Stevenson was adopted as the candidate for Banff and Buchan, ahead of the by-election on 7 June 2001 that was triggered by Alex Salmond's resignation from the seat to concentrate on Westminster politics. Stevenson was elected with a majority of 8,500 votes over the Conservative candidate.

He made his maiden speech on the European Union's Common Fisheries Policy on 14 June 2001.

He was re-elected to the constituency in the 2003 elections.

University Challenge
In 2004 he was a member of the Scottish Parliament team in the TV general knowledge programme, University Challenge – The Professionals. He and fellow team members Richard Baker (Labour), Robin Harper (Green), Jamie Stone (Lib Dem) who was captain, beat a Welsh Assembly team by 110 points to 75.

Responsibilities
In opposition he was Deputy Party Spokesperson on Health until September 2004, then becoming Deputy Party Spokesperson on Justice with responsibility for Prisons and Drugs policy. He was Convenor of the SNP Group in the Scottish Parliament and Deputy Convenor of the Parliament's Justice 1 Committee. In addition he ended Session 2 as a substitute member of the Parliament's Health Committee and Deputy Convenor of the Parliament's Cross Party Group on Visual Impairment.

Records
By the end of Parliament's second session on 2 April 2007 he had made 284 speeches in the Scottish Parliament and was thus the most prolific speaker since the Parliament's being re-convened in 1999. By the end of Session 3 in March 2011, he had made 406 speeches and retained the position of "most prolific parliamentary speaker". He reached his 500th speech on the Tribunals (Scotland) Bill, on 7 November 2013.

He can, arguably, hold the record for the longest speech in Parliament. He commenced a speech on International Suicide Prevention Week at 17:21 on Wednesday, 7 September 2004 and completed it at 17:12 on Thursday, 8 September 2004 nearly 24 hours later. However this was due to the failure of the Parliament's sound system just after he started to speak.

On 12 June 2015, he became the first Member of the Scottish Parliament to have made 600 speeches.

In Government
In the 2007 Scottish General election on 3 May, he was returned with a majority of 10,530, the largest in Scotland, over the Scottish Conservative Party candidate. The SNP formed a minority government and on 17 May Stevenson was appointed the Minister for Transport, Infrastructure and Climate Change. This appointment covered: the land use planning system, climate change, building standards, transport policy and delivery, public transport, road, rail services, canals, harbours, air and ferry services, Scottish Water.

As Minister, Stevenson piloted the SNP Government's first Bill, Abolition of Bridge Tolls (Scotland) Bill., which received royal assent on 24 January 2008, becoming the Abolition of Bridge Tolls (Scotland) Act 2008. At the end of May the Scottish Executive approved The Port of Cairnryan Harbour Empowerment Order 2007 and with this Stevenson became the first SNP Minister to sign a piece of legislation. He also brought forward the SNP's first Legislative Consent Motion, previously known as Sewel Motions, on the subject of the UK Climate Change Bill. He was also the first SNP Minister to lose a vote in Parliament on the subject of the Edinburgh Trams project.

As the Minister for Transport, he was involved with the progressing the legislation for the Forth Replacement Crossing, continuing a family association with Firth of Forth infrastructure projects. His great uncle, Sir Alexander Stevenson, was Chairman of the Forth Road Bridge Campaign Committee in the 1930s; the Forth Road Bridge opened in 1964.

In March 2009 Stevenson apologised for the use of an "intemperate word" in Parliament when he said the word "bollocks" in an off-mic remark in response to sedentary remarks by Liberal Democrat MSP Mike Rumbles on the relationship between Scottish ministers and officials at Transport Scotland.

After an unusually heavy snowfall in December 2010 caught authorities by surprise and left thousands of motorists stranded overnight on major highways, Stevenson called the government's response "first class" and refused to apologise; anger over the lack of preparedness and over his initial response made his position untenable, and he resigned on 11 December.

2011 Scottish Parliament election
In the 2011 Scottish Parliament election, under the re-drawn constituency boundaries, Stevenson was elected as the SNP member for the new seat of Banffshire and Buchan Coast; 16,812 votes cast for him was 67.24% of the total, the highest share of votes cast out of all the constituency elections for the Scottish Parliament in 2011.

Under the Second Salmond government Stevenson returned to a ministerial position, appointed as Minister for Environment and Climate Change on 20 May 2011. His ministerial role ended with the re-shuffle announced on 5 September 2012, when he was replaced as Minister by Paul Wheelhouse MSP.

2016 Scottish Parliament election
In the 2016 Scottish Parliament election, Stevenson was re-elected as member for Banffshire and Buchan Coast at the 2016 election with a reduced majority of 6,583.

2021 Scottish Parliament election
Stevenson did not stand for election in 2021, having previously announced his retirement.

Karen Adam was elected as his successor with a majority of 772.

Stevenson was elected National Secretary of the SNP on 30 November 2020 and served for a year.

In October 2022, Stevenson returned to University to study for an MSc in Genealogical, Palaeographic and Heraldic Studies at the University of Strathclyde.

title: Stewart Stevenson


text:
No Idea Records

No Idea Records is an American independent record label based in Gainesville, Florida which focuses on punk rock and its sub-styles and produces both vinyl records and compact discs.

No Idea Records started not as a record label, but as a zine in 1985, published independently by Var Thelin and Ken Coffelt and some friends of theirs from high school. By the seventh issue in 1989, Var was running the zine with Sarah Dyer and other contributors and collaborators. Starting with the sixth edition, the No Idea zine included 7-inch records with each issue. The first featured a local Gainesville band called Doldrums, and the second was a split 7-inch, one side of which belonged to later Bay Area legends Crimpshrine, a major influence on the musical style which dominates No Idea to the present.

Since its beginnings, No Idea has remained a foundation of the Gainesville punk scene and is considered by many to have spawned its very own style of punk, sometimes half-jokingly referred to as "beard punk" or "beardcore" due to the large proportion of members in bands having beards, or more commonly as variations of emo, post-hardcore and pop punk. The label has also worked with grindcore and metalcore bands like Left for Dead, Acrid, Bombs of Death, Crucible and The Swarm aka Knee Deep in the Dead.

Artists

 * Against Me!
 * Alkaline Trio
 * Ampere
 * Assholeparade
 * Atom & His Package
 * Bridge and Tunnel
 * Cheap Girls
 * Chuck Ragan
 * Coalesce
 * Combatwoundedveteran
 * Dead Bars
 * Defiance, Ohio
 * Dillinger Four
 * The Draft
 * The Ergs!
 * Floor
 * Good Luck
 * Hot Water Music
 * I Hate Myself
 * J Church
 * Jawbreaker
 * Latterman
 * Leatherface
 * Lemuria
 * Less Than Jake
 * Me First and the Gimme Gimmes
 * Moonraker
 * Mustard Plug
 * New Mexican Disaster Squad
 * No Friends
 * Off With Their Heads
 * Paul Baribeau
 * Planes Mistaken for Stars
 * Radon
 * Rehasher
 * Riverboat Gamblers
 * Rumbleseat
 * Samiam
 * Seaweed
 * Shores
 * Small Brown Bike
 * Strike Anywhere
 * Sunshine State
 * This Bike Is a Pipe Bomb
 * Tomorrow
 * Twelve Hour Turn
 * Western Addiction
 * A Wilhelm Scream
 * Worn in Red

Compilations

 * V/A – Back to Donut!
 * V/A – Bread: The Edible Napkin
 * V/A – Down in Front
 * V/A – Read Army Faction
 * V/A – No Idea 100: Redefiling Music
 * V/A – Tour Diary
 * V/A – Sight and Sound: The History of the Future
 * V/A – The Shape of Flakes to Come

title: No Idea Records


text:
Man, Ivory Coast

Man is a city in western Ivory Coast. It is the seat of both Montagnes District and Tonkpi Region. It is also a commune and the seat of and a sub-prefecture of Man Department. In the 2014 census, the city had a population of 149,041, making it the eighth-largest city in the country.

History
In November 2002, during conflict between government and rebel forces, the former rebel group Mouvement patriotique de Côte d'Ivoire (MPCI) held Man and the towns of Danané, Toulepleu, and Bloléquin.

Geography
Man is part of Montagnes District and is an important market town lying between mountains, including Mount Toura and Mount Tonkoui (the two highest in the nation), and La Dent de Man, popular with hikers and most recently, rock climbers.

Climate
Man has a tropical savanna climate (Köppen Aw).

Economy
Man is an agricultural region, with many cocoa, rice, coffee, cassava, banana (plantain), and soybean plantations. The area is the largest producer of coffee in the Ivory Coast. It is the home of a UNICAFÉ (The National Coffee Manufacturer of Côte d'Ivoire) factory, and Nestlé operates several coffee plants in the area. The central market of Man buys and sells a large variety of fabrics (Yacouba traditional clothes, so-called Boubou Yacouba), and an endless collection of Dan (Yacouba) masks.

The best known hotels are the "Hotel les Casacades", "Tanhotel", and "Hotel Beau Sejour". Those hotels are well-equipped with satellite TV, swimming pools, phone lines, and good service is provided.

Landmarks
The city is the home of the Roman Catholic Diocese of Man, with its cathedral at the Cathédrale Saint-Michel.

The main attractions around Man are La Cascade waterfall, a large rocky outcrop which overlooks the town known as the La Dent de Man (the tooth of Man), and the monkey forest. La Cascade waterfall is situated at the west side of the town, and is surrounded by a bamboo forest, which is the habitat of many colorful dragonflies and a wide variety of butterflies. The monkey forest is a wooded area in which monkeys live; for a small fee tourists can persuade local residents to call the monkeys out of their forest habitat.

Villages
The twenty nine villages of the sub-prefecture of Man and their population in 2014 are: • # Bantégouin (586)

• # Bigouin (1 091)

• # Blolé (2 784)

• # Botongouiné (1 131)

• # Dainé 2 (692)

• # Dompleu (2 622)

• # Gagouin (511)

• # Godégouin (569)

• # Gouakpalé (546)

• # Gouimpleu 1 (971)

• # Guianlé (1 421)

• # Kassiapleu (1 148)

• # Kpangouin 1 (1 018)

• # Kpangouin 2 (1 409)

• # Krikouma (1 205)

• # Man (148 945)

• # Petit Gbèpleu (648)

• # Seupleu (849)

• # Voungoué (1 764)

• # Yébégouin (324)

• # Zadépleu (463)

• # Zélé (3 042)

• # Biélé (1 533)

• # Gbatongouin (1 205)

• # Gueupleu (979)

• # Kiélé (2 918)

• # Lamapleu (1 132)

• # Oulédépleu (718)

• # Zérégouin (647)

title: Man, Ivory Coast


text:
Give and take

Give and take may refer to:
 * Give and Take (magazine), a Christadelphian Bible magazine aimed at 7- to 11-year-old Sunday School children
 * Give and Take (Smith), a 2005 public artwork on the campus of Indiana University – Purdue University Indianapolis
 * Give and Take (John Lindberg album), 1982
 * Give and Take (The Dynamic Superiors album), 1977
 * Give and Take (Mike Stern album), 1997
 * Give and Take (film), a 1928 silent film
 * Give and Take, a 1978 album by UK psychedelic/space rock band Here & Now
 * "Give and Take" (Red Dwarf), a 2016 episode of the TV series
 * "Give & Take", song by Netsky from the album 2, 2012
 * Give-n-Take, American television game show, 1975
 * Give and Take: A Revolutionary Approach to Success, book by Adam Grant, 2013
 * Give and Take Live, album by Zonke, 2013
 * Quid pro quo, a favor for a favor

title: Give and take


text:
Talk:No Idea Records

Ghost Mice not on No Idea?
Ghost Mice has never released a recording on No Idea Records. Their label Plan-it-X Records has a distribution deal with No Idea.

Should this also be listed on list of independent record labels? -- Ortonmc 18:56, 18 Nov 2003 (UTC)

Ghost Mice
Ghost Mice have released an LP entitled Europe on No Idea. It came out August of 2006.

title: Talk:No Idea Records