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Complete lattice

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A conditionally complete lattice is one that satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete.

Complete lattices appear in many applications in mathematics and computer science. Both order theory and universal algebra study them as a special instance of lattices.

Complete lattices must not be confused with complete partial orders (cpos), a more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales).

Formal definition
A complete lattice is a partially ordered set (L, ≤) such that every subset A of L has both a greatest lower bound (the infimum, also called the meet) and a least upper bound (the supremum, also called the join) in (L, ≤).

The meet is denoted by $$\bigwedge A$$, and the join by $$\bigvee A$$.

In the special case where A is the empty set, the meet of A is the greatest element of L. Likewise, the join of the empty set is the least element of L. Then, complete lattices form a special class of bounded lattices.

Complete sublattices
A sublattice M of a complete lattice L is called a complete sublattice of L if for every subset A of M the elements $$\bigwedge A$$ and $$\bigvee A$$, as defined in L, are actually in M.

If the above requirement is lessened to require only non-empty meet and joins to be in M, the sublattice M is called a closed sublattice of L.

Complete semilattices
The terms complete meet-semilattice or complete join-semilattice are another way to refer to complete lattices since arbitrary meets can be expressed in terms of arbitrary joins and vice versa (for details, see completeness).

Another usage of "complete meet-semilattice" refers to a meet-semilattice that is bounded complete and a complete partial order. This concept is arguably the "most complete" notion of a meet-semilattice that is not yet a lattice (in fact, only the top element may be missing).

See semilattices for further discussion between both definitions.

Conditionally Complete Lattices
A lattice is said to be "conditionally complete" if it satisfies either or both of the following properties:

 * Any subset bounded above has a least upper bound.
 * Any subset bounded below has a greatest lower bound.

Examples

 * Any non-empty finite lattice is trivially complete.
 * The power set of a given set is ordered by inclusion. The supremum is given by the union and the infimum by the intersection of subsets.
 * The unit interval [0,1] and the extended real number line, with the familiar total order and the ordinary suprema and infima. Indeed, a completely ordered set (with its order topology) is compact as a topological space if it is complete as a lattice.
 * The non-negative integers, ordered by divisibility. The least element of this lattice is the number 1, since it divides any other number. Perhaps surprisingly, the greatest element is 0, because it can be divided by any other number. The supremum of finite sets is given by the least common multiple and the infimum by the greatest common divisor. For infinite sets, the supremum will always be 0 while the infimum can well be greater than 1. For example, the set of all even numbers has 2 as the greatest common divisor. If 0 is removed from this structure it remains a lattice but ceases to be complete.
 * The subgroups of any given group under inclusion. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union itself.) If e is the identity of G, then the trivial group {e} is the minimum subgroup of G, while the maximum subgroup is the group G itself.
 * The submodules of a module, ordered by inclusion. The supremum is given by the sum of submodules and the infimum by the intersection.
 * The ideals of a ring, ordered by inclusion. The supremum is given by the sum of ideals and the infimum by the intersection.
 * The open sets of a topological space, ordered by inclusion. The supremum is given by the union of open sets and the infimum by the interior of the intersection.
 * The convex subsets of a real or complex vector space, ordered by inclusion. The infimum is given by the intersection of convex sets and the supremum by the convex hull of the union.
 * The topologies on a set, ordered by inclusion. The infimum is given by the intersection of topologies, and the supremum by the topology generated by the union of topologies.
 * The lattice of all transitive relations on a set.
 * The lattice of all sub-multisets of a multiset.
 * The lattice of all equivalence relations on a set; the equivalence relation ~ is considered to be smaller (or "finer") than ≈ if $$x \sim y $$ always implies $$x \approx y$$.
 * The lattice of self-adjoint projections (also known as orthogonal projections) of a von Neumann algebra.

Locally finite complete lattices
A complete lattice L is said to be locally finite if the supremum of any infinite subset is equal to 1, or equivalently, the set $$\{y \in L ~|~ y \le x\} \,\!$$ is finite for any $$1 \ne x \in L$$. The lattice (N, |) is locally finite. In this lattice, the element generally denoted "0" is actually 1 and vice versa.

Morphisms of complete lattices
The traditional morphisms between complete lattices are the complete homomorphisms (or complete lattice homomorphisms). These are characterized as functions that preserve all joins and all meets. Explicitly, this means that a function f: L→M between two complete lattices L and M is a complete homomorphism if

 * $$f\left(\bigwedge A\right) = \bigwedge\{f(a)\mid a\in A\}$$ and
 * $$f\left(\bigvee A\right) = \bigvee\{f(a)\mid a\in A\}$$,

for all subsets A of L. Such functions are automatically monotonic, but the condition of being a complete homomorphism is in fact much more specific. For this reason, it can be useful to consider weaker notions of morphisms, that are only required to preserve all joins (giving a category Sup) or all meets (giving a category Inf), which are indeed inequivalent conditions. This notion may be considered as a homomorphism of complete meet-semilattices or complete join-semilattices, respectively.

Galois connections and adjoints
Furthermore, morphisms that preserve all joins are equivalently characterized as the lower adjoint part of a unique Galois connection. For any pair of preorders P and Q, these are given by pairs of monotone functions f and g such that

 * $$f(x) \leq y \iff x \leq g(y)$$

where f is called the lower adjoint and g is called the upper adjoint. By the adjoint functor theorem, a monotone map between any pair of preorders preserves all joins if and only if it is a lower adjoint, and preserves all meets if and only if it is an upper adjoint.

As such, each join-preserving morphism determines a unique upper adjoint in the inverse direction that preserves all meets. Hence, considering complete lattices with complete semilattice morphisms boils down to considering Galois connections as morphisms. This also yields the insight that the introduced morphisms do basically describe just two different categories of complete lattices: one with complete homomorphisms and one with meet-preserving functions (upper adjoints), dual to the one with join-preserving mappings (lower adjoints).

A particularly important special case is for lattices of subsets $P(X)$ and $P(Y)$ and a function from X to Y. In this case, the direct image and inverse image maps between the power sets are upper and lower adjoints to each other, respectively.

Free "complete semilattices"
The construction of free objects depends on the chosen class of morphisms. Functions that preserve all joins (i.e. lower adjoints of Galois connections) are called free complete join-semilattices.

The standard definition from universal algebra states a free complete lattice over a generating set $$S$$ is a complete lattice $$L$$ together with a function $$i: S \rightarrow L$$, such that any function $$f$$ from $$S$$ to the underlying set of some complete lattice M can be factored uniquely through a morphism $$f^ \circ$$ from $$L$$ to $$M$$. Stated differently, for every element $$s$$ of $$S$$ it can be found that $$f(s) = f^\circ(i(s))$$ and that $$f^\circ$$ is the only morphism with this property. Hence, there is a functor from the category of sets and functions to the category of complete lattices and join-preserving functions which is left adjoint to the forgetful functor from complete lattices to their underlying sets.

Free complete lattices can thus be constructed such that the complete lattice generated by some set S is just the powerset $$2^S$$, the set of all subsets of S ordered by subset inclusion. The required unit $$i: S \rightarrow 2^S$$ maps any element s of S to the singleton set $${s}$$. Given a mapping $$f$$ as above, the function $$f^\circ : 2^S \rightarrow M$$ is defined by
 * $$f^\circ (X) = \bigvee \{ f(s) | s \in X \}$$.

Then $$f^\circ$$ transforms unions into suprema and thus preserves joins.

These considerations also yield a free construction for morphisms that preserve meets instead of joins (i.e. upper adjoints of Galois connections). The above can be dualized: free objects are given as powersets ordered by reverse inclusion, such that set union provides the meet operation, and the function $$f^\circ$$ is defined in terms of meets instead of joins. The result of this construction is known as a free complete meet-semilattice. It can be noted that these free constructions extend those that are used to obtain free semilattices, where finite sets need to be considered.

Free complete lattices
The situation for complete lattices with complete homomorphisms is more intricate. In fact, free complete lattices generally do not exist. Of course, one can formulate a word problem similar to the one for the case of lattices, but the collection of all possible words (or "terms") in this case would be a proper class, because arbitrary meets and joins comprise operations for argument sets of every cardinality.

This property in itself is not a problem: as the case of free complete semilattices above shows, it can well be that the solution of the word problem leaves only a set of equivalence classes. In other words, it is possible that the proper classes of all terms have the same meaning and are thus identified in the free construction. However, the equivalence classes for the word problem of complete lattices are "too small," such that the free complete lattice would still be a proper class, which is not allowed.

Now, one might still hope that there are some useful cases where the set of generators is sufficiently small for a free, complete lattice to exist. Unfortunately, the size limit is very low, and we have the following theorem:

 * The free complete lattice on three generators does not exist; it is a proper class.

A proof of this statement is given by Johnstone. The original argument is attributed to Alfred W. Hales; see also the article on free lattices.

Completion
If a complete lattice is freely generated from a given poset used in place of the set of generators considered above, then one speaks of a completion of the poset. The definition of the result of this operation is similar to the above definition of free objects, where "sets" and "functions" are replaced by "posets" and "monotone mappings". Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that are left adjoint to the forgetful functor in the converse direction.

As long as one considers meet- or join-preserving functions as morphisms, this can easily be achieved through the so-called Dedekind–MacNeille completion. For this process, elements of the poset are mapped to (Dedekind-) cuts, which can then be mapped to the underlying posets of arbitrary complete lattices in much the same way as done for sets and free complete (semi-) lattices above.

The aforementioned result that free complete lattices do not exist entails that an according free construction from a poset is not possible either. This is easily seen by considering posets with a discrete order, where every element only relates to itself. These are exactly the free posets on an underlying set. Would there be a free construction of complete lattices from posets, then both constructions could be composed, which contradicts the negative result above.

Representation
G. Birkhoff's book Lattice Theory contains a very useful representation method. It associates a complete lattice to any binary relation between two sets by constructing a Galois connection from the relation, which then leads to two dually isomorphic closure systems. Closure systems are intersection-closed families of sets. When ordered by the subset relation ⊆, they are complete lattices.

A special instance of Birkhoff's construction starts from an arbitrary poset (P,≤) and constructs the Galois connection from the order relation ≤ between P and itself. The resulting complete lattice is the Dedekind-MacNeille completion. When this completion is applied to a poset that already is a complete lattice, then the result is isomorphic to the original one. Thus, we immediately find that every complete lattice is represented by Birkhoff's method, up to isomorphism.

The construction is utilized in formal concept analysis, where one represents real-word data by binary relations (called formal contexts) and uses the associated complete lattices (called concept lattices) for data analysis. The mathematics behind formal concept analysis therefore is the theory of complete lattices.

Another representation is obtained as follows: A subset of a complete lattice is itself a complete lattice (when ordered with the induced order) if and only if it is the image of an increasing and idempotent (but not necessarily extensive) self-map. The identity mapping has these two properties. Thus all complete lattices occur.

Further results
Besides the previous representation results, there are some other statements that can be made about complete lattices, or that take a particularly simple form in this case. An example is the Knaster–Tarski theorem, which states that the set of fixed points of a monotone function on a complete lattice is again a complete lattice. This is easily seen to be a generalization of the above observation about the images of increasing and idempotent functions.

title: Complete lattice


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Trapani

Trapani (, ; Tràpani ) is a city and municipality (comune) on the west coast of Sicily, in Italy. It is the capital of the Province of Trapani. Founded by Elymians, the city is still an important fishing port and the main gateway to the nearby Egadi Islands.

History
Drepana was founded by the Elymians to serve as the port of the nearby city of Eryx (present-day Erice), which overlooks it from Monte Erice. The city sits on a low-lying promontory jutting out into the Mediterranean Sea.

It was originally named Drépanon from the Greek word for "sickle", because of the curving shape of its harbour. Carthage seized control of the city in 260BC, subsequently making it an important naval base, but ceded it to Rome in 241BC following the Battle of the Aegates in the First Punic War.

Two ancient legends relate supposed mythical origins for the city. In the first legend, Trapani stemmed from the sickle which fell from the hands of the goddess Demeter while she was seeking for her daughter Persephone, who had been kidnapped by Hades. The second myth features Kronos, who eviscerated his father Ouranos, god of the sky, with a sickle which, falling into the sea, created the city. In ancient times, Saturn (i.e., the Punic god Baal Hammon) was the patron god of Trapani. Today, Saturn's statue stands in a piazza in the centre of the city.

After its Roman, Vandal, Ostrogoth, Byzantine and (from 827) Arab conquests, Trapani was taken by the Normans of Roger I in 1077, flourishing under their dominations and having also a role in the Crusades as one of the most important ports in the Mediterranean Sea. During the first half of the 15th century, there was a community of Maltese people living in Trapani.

In the 17th century, the city decayed due to revolts, plagues, and famines, but in the following century, it grew from 16,000 to 30,000 inhabitants; commerce remained of local importance, while its military position in the Kingdom of Naples remained notable.

The city was badly damaged during World War II, when it was subjected to intense Allied bombardments. It has grown greatly since the end of the war, sprawling out virtually to the foot of Monte San Giuliano. Tourism has grown in recent years due to the city's proximity to popular destinations such as Erice, Segesta, and the Egadi Islands.

Geography
The comune of Trapani consists of two discontiguous parts separated by the comune of Paceco. The northern part includes much of the city and some rural area; the much larger southern part includes the area of Marausa, half of Trapani-Birgi Airport and a large rural area. The comune does not include the north-eastern suburbs of the urban area, such as Casa Santa, which are part of the comune of Erice. The comune of Trapani has a population of 70,000 but the entire urban, including those parts in the comune of Erice, has over 90,000 residents.

Economy
Much of Trapani's economy still depends on the sea and fishing and canning are the main local industries. Coral is also an important export, along with salt, marble, and marsala wine. The nearby coast is lined with numerous saltworks formed by the evaporation of seawater situated majestically along the coast road between Trapani and Marsala.

The city is also an important ferry port, with links to the Egadi Islands, Pantelleria, Sardinia, France and Tunisia. It also has its own airport, the Trapani-Birgi Airport.

Fishing
Trapani is one of the traditional locations of the mattanza tuna fishing technique, alongside: (San Giuliano, San Cusumano, Isola di Formica, Favignana, Bonagia, San Vito Lo Capo, Scopello, Capo Granitola). Today, this technique is forbidden but the fishing port is very active and hosts 142 small and medium fishing boats, for a total of 2805 GRT (gross tonnage).

The old fish market, renovated in 1998, is now used for cultural events and a new one, large and modern, more functional to fishing activities has been located near the port. It represents the only market in the Province and its recent restructuring, with European funds, places it at the forefront in the national level both in terms of marketing and product traceability.

Coral processing
Between the fifteenth and sixteenth centuries Trapani fishermen began to practice coral fishing, and coral craftmen started to develop its processing succeeding throughout the Mediterranean. A network of prestigious commissions was consolidated throughout Europe and it was thus possible to produce ever richer and more elaborate works. Today, however, fishing has almost completely disappeared, while coral processing is limited to few craftsmen.

Saltworks
Windmills and saltworks are evidence of industrial archeology. Saltworks are located in the area of Natural Reserve of Saline di Trapani and Paceco managed by the WWF and characterized by a remarkable flora and fauna. Thanks to the protection guaranteed by the Reserve, the activity of the saltworkers and the production of salt have increased, favoring the return and reproduction of dozens of species of migratory birds, including the pink flamingo.

Culture
The old city of Trapani dates from the later medieval or early modern periods; there are no more remains of the ancient city and many of the city's historic buildings are designed in the Baroque style.
 * The Church of Sant'Agostino (14th century)
 * The Church of Santa Maria di Gesù (15th–16th centuries)
 * Basilica-Sanctuary of Maria Santissima Annunziata (also called "Madonna di Trapani") originally built in 1315–1332 and rebuilt in 1760. It houses Museo regionale Agostino Pepoli and a marble statue of the Madonna of Trapani, which might be attributed to the work of Nino Pisano.
 * Fontana di Tritone ("Triton's Fountain")
 * The Baroque Palazzo della Giudecca  or Casa Ciambra.
 * The cathedral (built in 1421, but restored in the 18th century by Giovanni Biagio Amico). It includes a painting of "Annunciation" attributed to Anthony van Dyck.
 * Church of Maria SS. dell'Intria, an example of Sicilian Baroque.
 * Church of Badia Nuova, a small Baroque church.
 * Castello di Terra, a ruined 12th-century castle, today police office.
 * Ligny Tower, a 17th-century watchtower housing Phreistory museum.
 * Regional Museum Agostino Pepoli - Located in the 14th-century Carmelite convent, adjacent to the Sanctuary of Basilica-Sanctuary of Maria Santissima Annunziata, it is one of the most important Sicilian museums. It houses acollection of decorative arts, sculptures (including works by the Gagini), cribs and coral jewelry, and an art gallery that includes, among others, paintings by Titian and Giacomo Balla.
 * Museum of Prehistory - It is housed inside the seventeenth-century Torre di Ligny, on the extreme western point of the city, and preserves important prehistoric evidence of human presence in the area, as well as finds (artifacts, amphorae, anchors, a Punic helmet) from the sea of Trapani.
 * Museum of Contemporary Art San Rocco - Housed inside Palazzo San Rocco, in the historic center.
 * DiArt, diocesan collection of permanent religious art, housed in the episcopal seminary of Raganzìli in Casa Santa locality.
 * Diocesan Museum, in the Church of Sant'Agostino
 * Optical Illusions Museum
 * Specus Corallii designed by architect Antonino Cardillo

Folklore

The city is renowned for its Easter related Holy Week activities and traditions, culminating between Good Friday and Holy Saturday in the Processione dei Misteri di Trapani, colloquially simply the Misteri di Trapani (in English the Procession of the Mysteries of Trapani or the Mysteries of Trapani), a day-long passion procession organized and sponsored by the city's guilds, featuring twenty floats of wood, canvas and glue sculptures, mostly from the 17th and 18th centuries, of individual scenes of the events of the Passion.

The Misteri are among the oldest continuously running religious events in Europe, having been played every Good Friday since before the Easter of 1612. Running for at least 16 continuous hours, but occasionally well beyond the 24 hours, they are the longest religious festival in Sicily and in Italy. Important also to the cult of the Madonna of Trapani.

The city gives its name to a variety of pesto – pesto alla trapanese – made using almonds instead of the traditional pine nuts in Ligurian pesto.

Transport
Trapani-Birgi Airport is a military-civil joint use airport (third for traffic on the island). Recently the airport has seen an increase of traffic thanks to low-cost carriers from all parts of Europe (i.e. London-Stansted and London-Luton, Paris Beauvais, Dublin, Bruxelles, Munich, Frankfurt, Eindhoven, Stockholm, Malta, Bratislava).

Sport
From September 28 to October 9, 2005, Trapani was the location of Acts 8 and 9 of the Louis Vuitton Cup. This sailing race featured, among other entrants, all the boats that took part in the 2007 America's Cup.

The town is also the base for the local football team Trapani Calcio. Founded in 1905, they are nicknamed the Granata (the Maroons) after their kit colour. In 2010, Trapani Calcio was admitted into the 2010–11 Lega Pro Seconda Divisione (formerly Serie C2), ending the club's 13-year absence from the professional ranks. Subsequently, it made debut in Serie B in the 2013–14 season. It currently plays in Serie B with the coach Fabrizio Castori.

Climate
Trapani has a hot-summer mediterranean climate with hot and dry summers coupled with moderately wet and mild winters. Summer lows are cooler than in other places of Sicily and Calabria, while at the same time remaining significantly warm for several months.

Twin towns and Sister cities
Trapani is twinned with:
 * 🇷🇴 Constanța, Romania
 * 🇫🇷 Les Sables-d'Olonne, France
 * 🇫🇷 Roquefort-les-Pins, France
 * 🇩🇪 Würselen, Germany

Identification as Ithaca
Samuel Butler claimed, following visits in Trapani, that the city and its neighboring islands is the inspiration for both Odysseus's home of Ithaca as described in Homer's Oddysey, and Scheria, where Odysseus met Nausicaa.

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

title: Trapani


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Wilfred Bion

Wilfred Ruprecht Bion DSO (8 September 1897 – 8 November 1979) was an influential English psychoanalyst, who became president of the British Psychoanalytical Society from 1962 to 1965.

Early life and military service
Bion was born in Mathura, North-Western Provinces, India, and educated at Bishop's Stortford College in England. After the outbreak of the First World War, he served in the Tank Corps as a tank commander in France, and was awarded both the Distinguished Service Order (DSO) (on 18 February 1918, for his actions at the Battle of Cambrai), and the Croix de Chevalier of the Légion d'honneur. He first entered the war zone on 26 June 1917, and was promoted to temporary lieutenant on 10 June 1918, and to acting captain on 22 March 1918, when he took command of a tank section, he retained the rank when he became second-in-command of a tank company on 19 October 1918, and relinquished it on 7 January 1919. He was demobilised on 1 September 1921, and was granted the rank of captain. The full citation for his DSO reads:

"Awarded the Distinguished Service Order.

[...]

T./2nd Lt, Wilfred Ruprecht Bion, Tank Corps.

For conspicuous gallantry, and devotion to duty. When in command of his tank in an attack he engaged a large number of enemy machine guns in strong positions, thus assisting the infantry to advance. When his tank was put out of action by a direct hit he occupied a section of trench with his men and machine guns and opened fire on the enemy. He moved about in the open, giving directions to other tanks when they arrived, and at one period fired a Lewis gun with great effect from the top of his tank. He also got a captured machine gun into action against the enemy, and when reinforcements arrived he took command of a company of infantry whose commander was killed. He showed magnificent courage and initiative in a most difficult situation."

"Bion's daughter, Parthenope...raises the question of just how (and how far) her father was shaped as an analyst by his wartime experiences...under[p]inning Bion's later concern with the coexistence of regressed or primitive proto-mental states alongside more sophisticated one".

Education and early career
After World War I, Bion studied history at The Queen's College, Oxford, earning a Bachelor of Arts degree in 1922, before studying medicine at University College London.

Initially attracted to London by the "strange new subject called psychoanalysis", he met and was impressed by Wilfred Trotter, an outstanding brain surgeon who published the famous Instincts of the Herd in Peace and War in 1916, based on the horrors of the First World War. This was to prove an important influence on Bion's interest in group behavior. Having qualified in medicine by means of the Conjoint Diploma (MRCS England, LRCP London) in 1930 Bion spent seven years in psychotherapeutic training at the Tavistock Clinic, an experience he regarded, in retrospect, as having had some limitations. It did, however, bring him into fruitful contact with Samuel Beckett. He wanted to train in Psychoanalysis and in 1938 he began a training analysis with John Rickman, but this was brought to an end by the advent of the Second World War.

Bion was recommissioned in the Royal Army Medical Corps as a lieutenant on 1 April 1940, and worked in a number of military hospitals including Northfield Military Hospital (Hollymoor Hospital, Birmingham) where he initiated the first Northfield Experiment. These ideas on the psychoanalysis of groups were then taken up and developed by others such as S. H. Foulkes, Rickman, Bridger, Main and Patrick De Mare. The entire group at Tavistock had in fact been taken into the army, and were working on new methods of treatment for psychiatric casualties (those suffering post-traumatic stress, or "shell shock" as it was then known.) Out of this his pioneering work in group dynamics, associated with the "Tavistock group", Bion's papers describing his work of the 1940s were compiled much later and appeared together in 1961 in his influential book, Experiences in Groups and other papers. It was less a guide for the therapy of individuals within or by the group, than an exploration of the processes set off by the complex experience of being in a group. The book quickly became a touchstone work for applications of group theory in a wide variety of fields.

In 1945, during the Second World War, Bion's wife Betty Jardine gave birth to a daughter, but Betty died a few days afterwards. His daughter, Parthenope, became a psychoanalyst in Italy, and often lectured and wrote about her father's work. Parthenope died, together with her 18-year-old daughter Patrizia, in a car crash in Italy in July 1998.

Later career
Returning to the Tavistock Clinic Bion chaired the Planning Committee that reorganized the Tavistock into the new Tavistock Institute of Human Relations, alongside a new Tavistock Clinic which was part of the newly launched National Health Service. As his interest in psychoanalysis increased, he underwent training analysis, between 1946 and 1952, with Melanie Klein. He met his second wife, Francesca, at the Tavistock in 1951. He joined a research group of Klein's students (including Hanna Segal and Herbert Rosenfeld), who were developing Klein's theory of the paranoid-schizoid and the depressive positions, for use in the analysis of patients with psychotic disorders. He produced a series of highly original and influential papers (collected as "Second Thoughts", 1967) on the analysis of schizophrenia, and the specifically cognitive, perceptual, and identity problems of such patients. To this he added a valuable final section called Commentary, showing how some of his views on clinical and theoretical matters had changed.

Bion's theories, which were always based in the phenomena of the analytic encounter, revealed both correspondences and expansions of core ideas from both Sigmund Freud and Melanie Klein. At one point, he attempted to understand thoughts and thinking from an 'algebraic', 'geometric' and 'mathematised' point of view, believing there to be too little precision in the existing vocabulary, a process culminating in "The Grid". Later he abandoned the complex, abstract applications of mathematics, and the Grid, and developed a more intuitive approach, epitomised in Attention and Interpretation (1970).

In 1968, Bion moved to Los Angeles, California, where he remained until 1977. During those years he mentored a number of psychoanalysts interested in Kleinian approaches, including James Gooch (psychoanalyst) and other founding members of the Psychoanalytic Center of California. Shortly before his death, he returned to Oxfordshire.

Reception and stature
Bion left a reputation which has grown steadily both in Britain and internationally. Some commentators consider that his writings are often gnomic and irritating, but never fail to stimulate. He defies categorisation as a follower of Klein or of Freud. While Bion is most well known outside of the psychoanalytic community for his work on group dynamics, the psychoanalytic conversation that explores his work is mainly concerned with his theory of thinking, and his model of the development of a capacity for thought.

Wilfred Bion was a potent and original contributor to psychoanalysis. He was one of the first to analyse patients in psychotic states using an unmodified analytic technique; he extended existing theories of projective processes and developed new conceptual tools. The degree of collaboration between Hanna Segal, Wilfred Bion and Herbert Rosenfeld in their work with psychotic patients during the late 1950s, and their discussions with Melanie Klein at the time, means that it is not always possible to distinguish their exact individual contributions to the developing theory of splitting, projective identification, unconscious phantasy and the use of countertransference. As Donald Meltzer (1979, 1981), Denis Carpy (1989, p. 287), and Michael Feldman (2009, pp. 33, 42) have pointed out, these three pioneering analysts not only sustained Klein's clinical and theoretical approach, but through an extension of the concept of projective identification and countertransference they deepened and expanded it. In Bion's clinical work and supervision the goal remains insightful understanding of psychic reality through a disciplined experiencing of the transference–countertransference, in a way that promotes the growth of the whole personality.

'Bion's ideas are highly unique', so that he 'remained larger than life to almost all who encountered him'. He has been considered by Neville Symington as possibly "the greatest psychoanalytic thinker...after Freud".

Bion's work has left a strong impression on a number of contemporary psychoanalytic thinkers, including Antonino Ferro, Thomas Ogden, or Elias Mallet da Rocha Barros.

There is some historical evidence to suggest that the idea of containment may have been suggested to Bion in the mid-1930s, by an encounter with C.G.Jung: Bion attended Jung's 1935 lectures at the Tavistock Clinic, in which Bion was an active participant (asking three questions of Jung about a range of aspects of Jung's thinking). The experience was described by James Grotstein, Bion's biographer and "one of Bion's most influential pupils", as having had a "dramatic impact" on Bion.

Group experiments
Bion performed a lot of group experiments when he was put in charge of the training wing of a military hospital. Besides observing the basic assumptions recurring in these groups, he also has observed some very interesting phenomena which he believed may well apply to society.

About his experiences in groups, Bion wrote: "Judged by ordinary standards of social intercourse, the performance of the group is almost devoid of intellectual content. Furthermore, if we note how assumptions pass unchallenged as statements of fact, and are accepted as such, it seems clear that critical judgment is almost entirely absent." He explained this observation by calling attention to the presence of unconscious motivations underlying group behavior. This observation agrees with Gustave Le Bon's findings about groups to which he mentioned in his book The Crowd.

Bion also claimed that "...what the individual says or does in a group illumines both his own personality and his view of the group; sometimes his contribution illumines one more than the other." This observation is related to the psychological phenomenon of projection.

Bion posited the presence of "anonymous" contributions to a group as the foundations for "a successful system of evasion and denial". A related phenomenon is that of deindividuation.

In Experiences in Groups, Bion asserted that whenever a group is formed, it seeks a leader to follow. Moreover, he claimed that in the absence of a formal leader who satisfies the group, the most mentally ill member of the group will be treated as an informal leader: "In its search for a leader the group finds a paranoid schizophrenic or malignant hysteric if possible; failing either of these, a psychopathic personality with delinquent trends will do; failing a psychopathic personality it will pick on the verbally facile high-grade defective."

Group dynamics—the "basic assumptions"
Wilfred Bion's observations about the role of group processes in group dynamics are set out in Experiences in Groups and Other Papers, written in the 1940s but compiled and published in 1961, where he refers to recurrent emotional states of groups as 'basic assumptions'. Bion argues that in every group, two groups are actually present: the work group, and the basic assumption group. The work group is that aspect of group functioning which has to do with the primary task of the group—what the group has formed to accomplish; will "keep the group anchored to a sophisticated and rational level of behaviour". The basic assumption group describes the tacit underlying assumptions on which the behaviour of the group is based. Bion specifically identified three basic assumptions: dependency, fight-flight, and pairing. When a group adopts any one of these basic assumptions, it interferes with the task the group is attempting to accomplish. Bion believed that interpretation by the therapist of this aspect of group dynamics would, whilst being resisted, also result in potential insight regarding effective, co-operative group work.

In dependency, the essential aim of the group is to attain security through, and have its members protected by, one individual. The basic assumption in this group culture seems to be that an external object exists whose function it is to provide security for the immature individual. The group members behave passively, and act as though the leader, by contrast, is omnipotent and omniscient. For example, the leader may pose a question only to be greeted with docile silence, as though he or she had not spoken at all. The leader may be idealized into a kind of god who can take care of his or her children, and some especially ambitious leaders may be susceptible to this role. Resentment at being dependent may eventually lead the group members to "take down" the leader, and then search for a new leader to repeat the process.

In the basic assumption of fight-flight, the group behaves as though it has met to preserve itself at all costs, and that this can only be done by running away from someone or fighting someone or something. In fight, the group may be characterized by aggressiveness and hostility; in flight, the group may chit-chat, tell stories, arrive late or any other activities that serve to avoid addressing the task at hand. The leader for this sort of group is one who can mobilize the group for attack, or lead it in flight.

The final basic assumption group, pairing, exists on the assumption that the group has met for the purpose of reproduction—the basic assumption that two people can be met together for only one purpose, and that a sexual one'. Two people, regardless the sex of either, carry out the work of the group through their continued interaction. The remaining group members listen eagerly and attentively with a sense of relief and hopeful anticipation.

Bion considered that "the three basic-assumption groups seem each in turn to be aggregates of individuals sharing out between them the characteristics of one character in the Oedipal situation". Behind the Oedipal level, however, Bion postulated the existence of still more primitive, part-object phantasies; and "the more disturbed the group, the more easily discernible are these primitive phantasies and mechanisms". Such phantasies would prove the main focus of Bion's interest after his second analysis.

Bion on thinking
"During the 1950s and 1960s, Bion transformed Melanie Klein's theories of infantile phantasy...into an epistemological "theory of thinking" of his own." Bion used as his starting point the phenomenology of the analytic hour, highlighting the two principles of "the emergence of truth and mental growth. The mind grows through exposure to truth." The foundation for both mental development and truth are, for Bion, emotional experience.

The evolution of emotional experience into the capacity for thought, and the potential derailment of this process, are the primary phenomena described in Bion's model. Through his hypothesized alpha and beta elements, Bion provides a language to help one think about what is occurring during the analytic hour. These tools are intended for use outside the hour in the clinician's reflective process. To attempt to apply his models during the analytic session violates the basic principle whereby "Bion had advocated starting every session 'without memory, desire or understanding'—his antidote to those intrusive influences that otherwise threaten to distort the analytic process."

Alpha elements, beta elements, and alpha function
Bion created a theory of thinking based on changing beta elements (unmetabolized psyche/soma/affective experience) into alpha elements (thoughts that can be thought by the thinker). Beta elements were seen as cognate to the underpinnings of the "basic assumptions" identified in his work with groups: "the fundamental anxieties that underlie the basic assumption group resistances were originally thought of as proto-mental phenomena...forerunners of Bion's later concept of beta-elements." They were equally conceptual developments from his work on projective identification—from the "minutely split 'particles'" Bion saw as expelled in pathological projective identification by the psychotic, who would then go on to "lodge them in the angry, so-called bizarre objects by which he feels persecuted and controlled". For "these raw bits of experience he called beta-elements...to be actively handled and made use of by the mind they must, through what Bion calls alpha-functions, become alpha-elements".

β elements, α elements and α function are elements that Bion (1963) hypothesizes. He does not consider β-elements, α- elements, nor α function to actually exist. The terms are instead tools for thinking about what is being observed. They are elements whose qualities remain unsaturated, meaning we cannot know the full extent or scope of their meaning, so they are intended as tools for thought rather than real things to be accepted at face value (1962, p. 3).

Bion took for granted that the infant requires a mind to help it tolerate and organize experience. For Bion, thoughts exist prior to the development of an apparatus for thinking. The apparatus for thinking, the capacity to have thoughts "has to be called into existence to cope with thoughts" (1967, p. 111). Thoughts exist prior to their realization. Thinking, the capacity to think the thoughts which already exist, develops through another mind providing α-function (1962, p. 83)—through the "container" role of maternal reverie.

To learn from experience alpha-function must operate on the awareness of the emotional experience; alpha–elements are produced from the impressions of the experience; these are thus made storable and available for dream thoughts and for unconscious waking thinking... If there are only beta-elements, which cannot be made unconscious, there can be no repression, suppression, or learning. (Bion, 1962, p. 8)

α-function works upon undigested facts, impressions, and sensations, that cannot be mentalized—beta-elements. α-function digests β-elements, making them available for thought (1962, pp. 6–7). Beta-elements are not amenable to use in dream thoughts but are suited for use in projective identification. They are influential in producing acting out. These are objects that can be evacuated or used for a kind of thinking that depends on manipulation of what are felt to be things in themselves as if to substitute such manipulations for words or ideas... Alpha-function transforms sense impressions into alpha-elements which resemble, and may in fact be identical with, the visual images with which we are familiar in dreams, namely, the elements that Freud regards as yielding their latent content when the analyst has interpreted them. Failure of alpha-function means the patient cannot dream and therefore cannot sleep. As alpha-function makes the sense impressions of the emotional experience available for conscious and dream—thought the patient who cannot dream cannot go to sleep and cannot wake up. (1962, pp. 6–7)

Bizarre object
Bizarre objects, according to Bion, are impressions of external objects which, by way of projective identification, form a "screen" that's imbued with characteristics of the subject's own personality; they form part of his interpretation of object relations theory. Bion saw psychotic attacks on the normal linking between objects as producing a fractured world, where the patient felt themselves surrounded by hostile bizarre objects—the by-products of the broken linkages. Such objects, with their superego components, blur the boundary of internal and external, and impose a kind of externalised moralism on their victims. They can also contain ego-functions that have been evacuated from the self as part of the defence against thinking, sensing, and coming to terms with reality: thus a man may feel watched by his telephone, or that the music player being listened to is in fact listening to him in turn.

Later developments
Hanna Segal considered bizarre objects more difficult to re-internalise than either good or bad objects due to their splintered state: grouped together in a mass or psychic gang, their threatening properties may contribute to agoraphobia.

Knowledge, love and hate
Successful application of alpha-function leads to "the capacity to tolerate the actual frustration involved in learning ("K") that [Bion] calls 'learning from experience. The opposite of knowledge "K" was what Bion termed "−K": "the process that strips, denudes, and devalues persons, experiences, and ideas."

Both K and −K interact for Bion with Love and Hate, as links within the analytic relationship. "The complexities of the emotional link, whether Love or Hate or Knowledge [L, H, and K – the Bionic relational triad]" produce ever-changing "atmospheric" effects in the analytic situation. The patient's focus may wish to be "on Love and Hate (L and H) rather than the knowledge (K) that is properly at stake in psychoanalytic inquiry."

For Bion, "knowledge is not a thing we have, but a link between ourselves and what we know ... K is being willing to know but not insisting on knowledge." By contrast, -K is "not just ignorance but the active avoidance of knowledge, or even the wish to destroy the capacity for it" – and "enacts what 'Attacks on Linking' identifies as hatred of emotion, hatred of reality, hatred of life itself."

Looking for the source of such hate (H), Bion notes in Learning from Experience that, "Inevitably one wonders at various points in the investigation why such a phenomenon as that represented by −K should exist. ... I shall consider one factor only – Envy. By this term I mean the phenomenon described by Melanie Klein in Envy and Gratitude" (1962, p. 96).

Reversible perspective and −K
"Reversible Perspective" was a term coined by Bion to illuminate "a peculiar and deadly form of analytic impasse which defends against psychic pain". It represents the clash of "two independently experienced views or phenomena whose meanings are incompatible". In Bion's own words, "Reversible perspective is evidence of pain; the patient reverses perspective so as to make a dynamic situation static."

As summarised by Etchegoyen, "Reversible perspective is an extreme case of rigidity of thought. ... As Bion says, what is most characteristic in such cases is the manifest accord and the latent discord." In clinical contexts, what may happen is that the analyst's "interpretation is accepted, but the premises have been rejected ... the actual specificity, the substance of the interpretation". Reversible perspective is an aspect of "the potential destruction and deformation of knowledge" – one of the attacks on linking of −K.

O: The ineffable
As his thought continued to develop, Bion came to use Negative Capability and the suspension of Memory and Desire in his work as an analyst, in order to investigate psychic reality - which he regarded as essentially 'non-sensuous' (1970). Following his 1965 book Transformations he had an increasing interest in what he termed the domain of "O" – the unknowable, or ultimate Truth. "In aesthetics, Bion has been described as a neo-Kantian for whom reality, or the thing-in-itself (O), cannot be known, only be "be-ed" (1965). What can be known is said by Bion to be in the realm of K, impinging through its sensory channels. If the observer can desist from "irritably reaching for fact and reason", and suspend the normal operation of the faculties of memory and apperception, what Bion called transformations in knowledge can permit an 'evolution' where transformations in K touch on transformations in Being (O). Bion believed such moments to feel both ominous and turbulent, threatening a loss of anchorage in everyday 'narrative' security.

Bion would speak of "an intense catastrophic emotional explosion O," which could only be known through its aftereffects. Where before he had privileged the domain of knowledge (K), now he would speak as well of "resistance to the shift from transformations involving K (knowledge) to transformations involving O ... resistance to the unknowable". Hence his injunctions to the analyst to eschew memory and desire, to "bring to bear a diminution of the 'light' – a penetrating beam of darkness; a reciprocal of the searchlight. If any object existed, however faint, it would show up very clearly". In stating this he was making connections to Freud, who in a letter to Lou Andreas Salome had referred to a mental counterpart of scotopic, "mole like vision", used to gain impressions of the Unconscious. He was also making links with the apophatic method used by contemplative thinkers such as St John of the Cross, a writer quoted many times by Bion. Bion was well aware that our perception and our attention often blind us to what genuinely and strikingly is new in every moment.

Reverie
Bion's concept of maternal "reverie" as the capacity to sense (and make sense of) what is going on inside the infant has been an important element in post-Kleinian thought: "Reverie is an act of faith in unconscious process ... essential to alpha-function'" It is considered the equivalent of Stern's attunement, or Winnicott's maternal preoccupation.

In therapy, the analyst's use of "reverie" is an important tool in his/her response to the patient's material: "It is this capacity for playing with a patient's images that Bion encouraged".

Late Bion
"For the later Bion, the psychoanalytic encounter was itself a site of turbulence, 'a mental space for further ideas which may yet be developed'." In his unorthodox quest to maintain such "mental space", Bion "spent the final years of his long and distinguished professional life [writing] a futuristic trilogy in which he is answerable to no one but himself, A Memoir of the Future."

If we accept that "Bion introduced a new form of pedagogy in his writings...[via] the density and non-linearity of his prose", it comes perhaps to a peak here in what he himself termed "a fictitious account of psychoanalysis including an artificially constructed dream ... science fiction". We may conclude at least that he achieved his stated goal therein: "To prevent someone who KNOWS from filling the empty space".

title: Wilfred Bion


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Gibellina

Gibellina (Sicilian: Jibbiddina, Arabic: "little mount" - جبل صغير) is a small city and comune in the Province of Trapani, the mountains of central Sicily, Italy. It was destroyed by the 1968 Belice earthquake.

The new city, Gibellina Nuova, was rebuilt some 11 km from the old one and it was designed by some of the most prominent artists and architects in Italy. They were summoned by Ludovico Corrao to provide works of art to the city in order to help build it up as an eccentric museum en plein air. One of them, the Italian sculptor Pietro Consagra, created a sculpture called Porta del Belice, or "Door to Belice", at the entrance. Consagra expressed on his deathbed a wish to be buried at Gibellina in July 2005.

The old town, now known as the Ruderi di Gibellina (as the ruins of the city are referred to), remained just as it was after the earthquake, like a ghost town until 1985. That year the Italian artist Alberto Burri began a project to cover the ruins in concrete, while preserving the streetscape. Known as Cretto di Burri, work on the project ceased in 1989, but was finally completed in 2015.

title: Gibellina


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Erice

Erice (Èrici [ˈɛːɾɪʃɪ]) is a historic town and comune in the province of Trapani, Sicily, southern Italy. It is a member of the I Borghi più belli d'Italia ("The most beautiful villages of Italy") association.

Geography
The main town of Erice is located on top of Mount Erice, at around 750 m above sea level, overlooking the city of Trapani, the low western coast towards Marsala, the dramatic Punta del Saraceno and Capo San Vito to the north-east, and the Aegadian Islands on Sicily's north-western coast. Casa Santa forms part of Erice at the base of Mount Erice, immediately adjacent to Trapani. A cable car joins the upper and lower parts of Erice.

The bordering municipalities are Buseto Palizzolo, Paceco, Trapani, Valderice and Custonaci. The hamlets (frazioni) are Ballata, Casa Santa, Crocefissello, Napola, Pizzolungo, Rigaletta, San Cusumano and Torretta.

History

The ancient Greek name of Erice was Eryx (Έρυξ in Greek), and its foundation was associated with the eponymous Greek hero Eryx. It was not a Greek colony, as the Phoenicians founded it, but was largely Hellenized. It was destroyed in the First Punic War by the Carthaginians, and from then on declined in importance.

Eryx was conquered by the Aghlabids in 831 and was renamed as Cebel Hamid (in Western sources Gebel Hamed, meaning Mountain of Hamid). It was ruled by the Arabs until the Norman conquest. In 1167 the Normans renamed it Monte San Giuliano, a name maintained until the 20th century.

20th Century
In 1934 the town changed its name from Monte San Giuliano to Erice. During the Second World War a Luftwaffe Operations Centre (associated with Zerstörergeschwader 26 and Jagdgeschwader 27 which operated out of the nearby Trapani–Milo Airport) was located on the slopes of Monte Erice from early 1943 until they were forced by Allied air raids to relocate. Following the Allied invasion of Sicily in 1943 troops of the 2nd Battalion of the 505th Infantry Regiment  Seventh Army under the command of Major Mark Alexander began on 22 July 1943 to climb up the side of Monte Erice in order to secure the town and its commanding position overlooking the surrounding countryside. As they did so they came under artillery fire from Italian forces stationed on the ramparts with one soldier being killed and another wounded. Realizing that the safest position was directly under the walls of town the lead troops took shelter there. Rather than waiting on artillery support to arrive the decision was then made for F Company to attack. However before the troops had come close enough to engage them the Italian forces signaled their surrender.

Main sights
In the northeastern portion of the city there are the remains of ancient Elymian and Phoenician walls (Cyclopean masonry) indicating different stages of settlement and occupation in antiquity.

There are two castles that remain in the city: Pepoli Castle, which dates from Saracen times, and the Castello di Venere ("Venus Castle"), dating from the Norman period, built on top of the ancient Temple of Venus, where Venus Ericina was worshipped. According to legend, the temple was founded by Aeneas. It was well known throughout the Mediterranean area in the ancient age, and an important cult was celebrated in it. In his book On the Nature of Animals, Aelian writes that animals chosen for sacrifice would voluntarily walk up to the altar to be killed.

A cable car (funivia) ran from 2005 to 2017, when it was closed due to a forest fire, from the outskirts of Trapani to the town of Erice. The cable car was rebuilt and reopened in June 2018.

Culture
Erice hosts scientific meetings at the Ettore Majorana center, organised by the controversial astrophysicist Antonino Zichichi. There is also an annual workshop on Molecular Gastronomy.

title: Erice


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Battle of Stirling Bridge

The Battle of Stirling Bridge (Blàr Drochaid Shruighlea) was fought during the First War of Scottish Independence. On 11 September 1297, the forces of Andrew Moray and William Wallace defeated the combined English forces of John de Warenne, 6th Earl of Surrey, and Hugh de Cressingham near Stirling, on the River Forth.

Background
In 1296, John de Warenne, 6th Earl of Surrey, defeated John Comyn, Earl of Buchan in the Battle of Dunbar. King John Balliol surrendered to King Edward I of England at Brechin on 10 July, and the Scottish landholders were made to acknowledge Edward's overlordship. In 1297, Moray initiated a revolt in northern Scotland and by the late summer, controlled Urquhart, Inverness, Elgin, Banff and Aberdeen. Wallace joined Moray in September near Dundee, and they marched to Stirling. Stirling, in the words of Stuart Reid, was "traditionally regarded as the key to Scotland." Meanwhile, Surrey had joined Cressingham in July and both had arrived at Stirling by 9 September 1297. By then, Moray and Wallace had already occupied Abbey Craig.

The main battle
Surrey was concerned with the number of Scots he faced, separated by a long causeway and narrow, wooden bridge, over the River Forth near Stirling Castle. Determining that he would be at a tactical disadvantage if he attempted to take his main force across there, he delayed crossing for several days to allow for negotiations and to reconnoiter the area. On 11 September, Surrey had sent James Stewart, and then two Dominican friars as emissaries to the Scots. According to Walter of Guisborough, Wallace reputedly responded with, "We are not here to make peace but to do battle to defend ourselves and liberate our kingdom. Let them come on and we shall prove this to their very beards."

Camped on Abbey Craig, the Scots dominated the soft flat ground north of the river. The English force of English, Welsh and Scots knights, bowmen and foot soldiers camped south of the river. Sir Richard Lundie, a Scots knight who joined the English after the Capitulation of Irvine, offered to outflank the enemy by leading a cavalry force over a ford two miles upstream, where sixty horsemen could cross at the same time. Hugh de Cressingham, King Edward's treasurer in Scotland, persuaded the Earl to reject that advice and order a direct attack across the bridge.

The small bridge was broad enough to let only two horsemen cross abreast but offered the safest river crossing, as the Forth widened to the east and the marshland of Flanders Moss lay to the west. The Scots waited as the English knights and infantry, led by Cressingham, with Sir Marmaduke Thweng and Sir Richard Waldegrave, began to make their slow progress across the bridge on the morning of 11 September. It would have taken several hours for the entire English army to cross.

Wallace and Moray waited, according to the Chronicle of Hemingburgh, until "as many of the enemy had come over as they believed they could overcome". When a substantial number of the troops had crossed (possibly about 2,000) the attack was ordered. The Scots spearmen came down from the high ground in rapid advance and fended off a charge by the English heavy cavalry and then counterattacked the English infantry. They gained control of the east side of the bridge and cut off the chance of English reinforcements to cross. Caught on the low ground in the loop of the river with no chance of relief or of retreat, most of the outnumbered English on the east side were probably killed. A few hundred may have escaped by swimming across the river. Marmaduke Thweng managed to fight his way back across the bridge with some of his men.

Surrey, who was left with a small contingent of archers, had stayed south of the river and was still in a strong position. The bulk of his army remained intact and he could have held the line of the Forth, denying the Scots a passage to the south, but his confidence was gone. After the escape of Sir Marmaduke Thweng, Surrey ordered the bridge to be destroyed, retreated towards Berwick, leaving the garrison at Stirling Castle isolated and abandoning the Lowlands to the rebels. James Stewart, the High Steward of Scotland, and Malcolm, Earl of Lennox, whose forces had been part of Surrey's army, observing the carnage to the north of the bridge, withdrew. Then the English supply train was attacked at The Pows, a wooded marshy area, by James Stewart and the other Scots lords, killing many of the fleeing soldiers.

The Stirling Bridge of that time is believed to have been about 180 yards upstream from the 15th-century stone bridge that now crosses the river. Four stone piers have been found underwater just north (56.12917°N, -3.93667°W) and at an angle to the extant 15th-century bridge, along with man-made stonework on one bank in line with the piers. The site of the fighting was along either side of an earthen causeway leading from the Abbey Craig, atop which the Wallace Monument is now, to the north end of the bridge. The battlefield has been inventoried and protected by Historic Scotland under the Scottish Historical Environment Policy of 2009.

Aftermath
Surrey left William de Warine and Sir Marmaduke Thweng in charge of Stirling Castle, as Surrey abandoned his army, and fled towards Berwick.

The contemporary English chronicler Walter of Guisborough recorded the English losses in the battle as 100 cavalry and 5,000 infantry killed. Scottish casualties in the battle are unrecorded, with the exception of Andrew Moray, who was mortally wounded during the battle, and was dead by November.

The Lanercost Chronicle records that Wallace had a broad strip of Cressingham's skin, "...taken from the head to the heel, to make therewith a baldrick for his sword."

The Scots proceeded to raid the south as far as Durham, England. Wallace was appointed "Guardian of the kingdom of Scotland and commander of its army." Yet, Edward was already planning another invasion of Scotland, which would lead to the Battle of Falkirk.

Popular culture
The exploits of Wallace were passed on to posterity mainly in the form of tales collected and recounted by the poet Blind Harry, the Minstrel (d. 1492), whose original, probably oral sources were never specified. Blind Harry was active some 200 years after the events described in his The Acts and Deeds of the Illustrious and Valiant Champion Sir William Wallace, c. 1470. The tales were designed to entertain the court of James IV (r. 1488–1513) and are undoubtedly a blend of fact and fiction. Like most of his other episodes, Blind Harry's account of the battle of Stirling Bridge is highly improbable, such as his use of figures of biblical magnitude for the size of the participating armies. Nevertheless, his highly dramatised and graphic account of the battle fed the imaginations of subsequent generations of Scottish schoolchildren. Here is his description:

On Saturday they [Moray and Wallace] rode on to the bridge, which was of good plain board, well made and jointed, having placed watches to see that none passed from the army. Taking a wright, the most able workman there, he [Wallace] ordered him to saw the plank in two at the mid streit [middle stretch], so that no-one might walk over it. He then nailed it up quickly with hinges, and dirtied it with clay, to cause it to appear that nothing had been done. The other end he so arranged that it should lie on three wooden rollers, which were so placed, that when one was out the rest would fall down. The wright, himself, he ordered to sit there underneath, in a cradle, bound on a beam, to loose the pin when Wallace let him know by blowing a horn when the time was come. No one in all the army should be allowed to blow but he himself.

Hugh Cressingham leads on the vanguard with twenty thousand likely men to see. Thirty thousand the Earl of Warren had, but he did then as wisdom did direct, all the first army being sent over before him. Some Scottish men, who well knew this manner of attack, bade Wallace sound, saying there were now enough. He hastened not, however, but steadily observed the advance until he saw Warren's force thickly crowd the bridge. Then from Jop he took the horn and blew loudly, and warned John the Wright, who thereupon struck out the roller with skill; when the pin was out, the rest of it fell down. Now arose an hideous outcry among the people, both horses and men, falling into the water. (...)

On foot and bearing a great sharp spear, Wallace went amongst the thickest of the press and aimed a stroke at Cressingham in his corslet, which was brightly polished. The sharp head of the spear pierced right through the plates and through his body, stabbing him beyond rescue; thus was that chieftain struck down to death. With the stroke Wallace bore down both man and horse.

The English army although ready for battle, lost heart when their chieftain was slain and many openly began to flee. Yet, worthy men abode in the place until ten thousand were slain. Then the remainder fled, not able to abide longer, seeking succour in many directions, some east, some west and some fled to the north. Seven thousand full at once floated in the Forth, plunged into the deep and drowned without mercy; none were left alive of all that fell army.

As well as the bridge ploy, Wallace's use of a spear appears to be a fictional element. A two-handed sword [claidheamh dà-làimh, in Gaelic, more commonly claidheamh-mòr or claymore meaning great sword], purporting to be Wallace's, which may contain original metal from his sword blade, was kept by the Scottish kings and is displayed as a relic in the Wallace Monument.

The potency of these tales can be gauged from the following statement by the poet Robert Burns, writing some three centuries after they were first related.

"The two first books I ever read in private, and which gave me more pleasure than any two books I ever read again, were The Life of Hannibal and The History of Sir William Wallace [a modernised version of Blind Harry by William Hamilton of Gilbertfield]. Hannibal gave my young ideas such a turn that I used to strut in raptures up and down after the recruiting drum and bagpipe, and wish myself tall enough that I could be a soldier; while the story of Wallace poured a Scottish prejudice into my veins which will boil along there till the flood-gates of life shut in eternal rest."

The Battle of Stirling Bridge is depicted in the 1995 film Braveheart, but it bears little resemblance to the real battle. Significantly, there is no bridge in the movie (due mainly to the difficulty of filming around the bridge itself), and it replaced the tactics of the battle with ones resembling the Battle of Bannockburn.

title: Battle of Stirling Bridge


text:
Apulia

Apulia, also known by its Italian name Puglia , is a region of Italy, located in the southern peninsular section of the country, bordering the Adriatic Sea to the east, the Strait of Otranto and Ionian Sea to the southeast and the Gulf of Taranto to the south. The region comprises 19345 sqkm, and its population is about four million people. It is bordered by the other Italian regions of Molise to the north, Campania to the west, and Basilicata to the southwest. The regional capital is Bari.

In ancient times, more precisely at the beginning of the first millennium BC, the region of Apulia was inhabited by the Iapygians, while during the 8th century BC its coastal areas were populated by ancient Greeks. Later, the region was conquered by the ancient Romans. It was then conquered by the Byzantines, followed by the Normans, the Aragonese and the Spanish. Subsequently, it became part of the Kingdom of the Two Sicilies, to then be annexed to the unified Kingdom of Italy after the Expedition of the Thousand.

Geography
Apulia's coastline is longer than that of any other mainland Italian region. In the north, the Gargano promontory extends out into the Adriatic sea like a "sperone" ("spur"), while in the south, the Salento peninsula forms the "tacco" ("heel") of Italy's boot. The highest peak in the region is Monte Cornacchia 1,152 m above sea level) within the Daunian Mountains, in the north along the Apennines.

It is home to two national parks, the Alta Murgia National Park and Gargano National Park.

Outside national parks in the North and West, most of Apulia particularly the Salento peninsula is geographically flat with only moderate hills.

The climate is typically Mediterranean with hot, dry and sunny summers and mild and rainy winters. Snowfall, especially on the coast is rare but has occurred as recently as January 2019 (following on from snow in March 2018 and January 2017). Apulia is among the hottest and driest regions of Italy in summer, with temperatures sometimes reaching and exceeding 40 C in Lecce and Foggia.

The coastal areas, particularly on the Adriatic Sea and in the southern Salento peninsula, are frequently exposed to winds of varying strengths and directions, strongly affecting local temperatures and conditions, sometimes within the same day. The Northerly Bora wind from the Adriatic Sea can lower temperatures, humidity and moderate summer heat while the Southerly Sirocco wind from North Africa can raise temperatures, and humidity and occasionally drop red dust from the Sahara Desert. On some days in spring and autumn/fall, it can be warm enough to swim in Gallipoli and Porto Cesareo on the Ionian coast while at the same time, cool winds warrant jackets and jumpers/sweaters in Monopoli and Otranto on the Adriatic coast.

The area between Otranto and Santa Maria di Leuca is part of the Regional Natural Coastal Park of "Costa Otranto — Santa Maria di Leuca e Bosco di Tricase" wanted by the Apulia Region in 2008. This territory has numerous natural and historical attractions such as Ciolo, which is a rocky cove. In 2010 the Ministry of Health declared 98.6% of the Apulian coasts suitable for bathing.

Antiquity
Human settlement in Apulia dates back to at least 250,000 years ago, as evidenced by the fossil remains of the Altamura Man, an archaic form of Neanderthal. There are numerous finds from the prehistoric era, including several menhir and dolmen.

Around the 1st millennium BC, the Iapygians settled in the territory with the tribes of the Daunians, the Peucetians and the Messapians, as well as the populations of the Calabri and the Sallentini (both settled in Salento); later, in the Hellenic era, the Magna Graecia colonies were quite numerous, especially in the southern part of the region, including the city of Taras, now Taranto.

During the second Samnite war (326 - 304 BC), the Roman army, in an attempt to provide relief to Lucera, besieged by the Samnites, suffered a serious defeat in the Battle of the Caudine Forks, in 321 BC. Rome soon understood the strategic importance of Apulia (corresponding only to the central-northern part of present-day Apulia, while the Salento Peninsula was known as Calabria in that period), but the occupation of the region, in the third century BC, was not easy, especially for the resistance of Tarentum and Brundisium. In 216 BC in Cannae the Roman army suffered its worst defeat against Hannibal's Carthage.

The Roman Province of Apulia and Calabria was then established, which also included Irpinia. With the construction of the Appian Way and, in the imperial era, of the Via Traiana along which cities such as Troia, Ordona, Gravina, Canosa, Ruvo and Bitonto prospered. The region occupied leading positions in the production of grain and oil, becoming the largest exporter of olive oil in the East.

Middle Ages and Renaissance period
At the Fall of the Western Roman Empire, Apulia also went through a long period of suffering. Heruli and Ostrogoths invaded the territory, but in the end it became the dominion of the Eastern Roman Empire, from the 6th to the 11th century. Bari became the capital of a territory extending up to present-day Basilicata and subjected to the authority of a Catepanate. With the advent of the Normans in the 11th century, Taranto became the capital of the Principality of Taranto, which extended across the entire Terra di Otranto.

In 1043 the Normans founded the county of Apulia which in 1059 merged into the vast County of Apulia and Calabria, whose borders progressively extended up to in Principality of Salerno. From 1130 it became part of the Kingdom of Sicily. In the 13th century the name Apulia was used by some authors to indicate the southern part of the Italian peninsula. During the Sicilian domination Apulia achieved great material and civil progress, which reached its peak with Frederick II, who was responsible for the construction of a series of secular and religious buildings, some of high artistic value, including Castel del Monte, near Andria. During this period Foggia became one of his residences. From 1282, following the separation of the island of Sicily from the rest of the south of the peninsula, Apulia was under the rule of the Kingdom of Naples, from that moment the power of the landowners began to take root in the territory.

Early modern period
From the middle of the 18th century the region of Apulia experienced a period of strong economic prosperity and excellent development of trade and agriculture. Between 1806 and 1815, during the Napoleonic era, provided the modernization of Apulia with the abolition of feudalism and judicial reforms until the return of the Bourbons and the birth of the Kingdom of the Two Sicilies. Liberal movements were formed throughout the region in 1820 with the spread of Freemasonry and Carbonari.

With the Kingdom of Italy established in 1861, Apulia was administratively divided into the provinces of Foggia, Bari and Lecce; to these were added in the twentieth century the provinces of Brindisi and Taranto. In the period following the unification of Italy, various brigand gangs arose, especially in Capitanata and Terra di Bari; among the major exponents are Michele Caruso, Antonio Angelo Del Sambro and Giuseppe Schiavone, the latter a faithful lieutenant of the Lucanian brigand leader Carmine Crocco.

Modern period
With the progressive decline of the latifundium, the ancient Apulian farms, properties of medium agricultural size, also decayed. During fascism, Apulia was affected by numerous land reclamations in vast areas and, following the post-war agrarian reform, the region enjoyed strong agricultural development. In the 1970s and 1980s the economy of the region moved from the primary sector to the tertiary one, with notable development coming from the tourism sector.

Economy
The region's contribution to Italy's gross value added was around 4.6% in 2000, while its population was 7% of the total. The per capita GDP is low compared to the national average and represents about 68.1% of the EU average.

The share of gross value added by the agricultural and services sectors was above the national average in 2000. The region has industries specialising in particular areas, including food processing and vehicles in Foggia; footwear and textiles in the Barletta province, and wood and furniture in the Murge area to the west.

Between 2007 and 2013 the economy of Apulia expanded more than that of the rest of southern Italy. Such growth, over several decades, is a severe challenge to the hydrogeological system.

Apulia's thriving economy is articulated into numerous sectors boasting several leading companies, but most of them produce materials or components, not finished goods:
 * Aerospace (Leonardo, Avio Aero, Sitael, Blackshape)
 * Automotive (Bosch, Magneti Marelli, Magna Gertrag, Bridgestone)
 * Mechanics (New Holland Construction, IVECO Motori, Isotta Fraschini Motori, MERMEC)
 * Furniture (Natuzzi)
 * Food and Beverage (De Carlo, Divella, Quarta Caffé)
 * Agriculture (Casillo Group, G.C. Partecipazioni)
 * Publishing (Laterza, Edizioni Dedalo)
 * Tourism (Nicolaus tour)
 * Logistics (GTS Rail)
 * I.C.T. (Exprivia)

In Taranto, there is the largest metallurgical work ILVA Acciaierie di Taranto (8,200 empl.) in Europe with a full iron and steel production cycle. It will be sold to Arcelor Mittal.

In Brindisi, there is a chemical industrial park with an ENI power station, ENI oil refinery, Syndial chemical factory, and Versalis chemical factory (basic chemistry, intermediates, polyethene, styrenics and elastomers). LyondellBasell polypropylene plant, Sanofi plant (antibiotics). Another ENI oil refinery is located in Taranto.

Unemployment
The unemployment rate stood at 14.1% in 2020.

Fishing and aquaculture
The port of Taranto hosts numerous fishing boats. The fleet is mainly made up of about 80 fishing boats, which do not exceed 10 gross tonnage and which practice trawling, while the remaining small-scale fishing boats operate with gillnets. The sea, rich and generous, is populated by dentex and sea bream, grouper, mullet and anchovies, shrimp and squid. Other significant fishing ports are Manfredonia, Molfetta, Mola di Bari, Monopoli, Gallipoli and Castro.

Today Taranto is the world's largest producer of farmed mussels: with 1,300 employees, around 30,000 tons of mussels are processed per year. Mussel farming has characterized the city's economy for centuries, making the mussel the gastronomic symbol of Taranto. It is said that the first mussel gardens in La Spezia, Pula, Olbia and Chioggia were established by mussel farmers who emigrated from this city. The workplace of the Taranto mussel farmers is the boat; every detail of the working method has improved over time.

10 m long structures made of wood or metal, called "pali" (piles), are attached to the seabed, to which ropes and nets are then attached, on which the mussels are grown. The mussels farmed here are particularly tasty and valued because they grow in a special environment, a mixture of salt seawater and karst freshwater. These special environmental conditions of the seas of Taranto are ideal not only for the mussels but also for the fish and crustaceans that find food and shelter between the piles. While there are around 18 submarine freshwater springs, called "Citri", in the Mar Piccolo, there is only one large one in the Mar Grande, which is called "Anello di San Cataldo" in honour of the patron saint of the city.

Agriculture
Agriculture plays a prominent economic role in Apulia. It is mainly intensive and modern agriculture that allows the region to be among the first in Italy for the production of many products:
 * durum wheat which is used to produce pasta
 * tomatoes
 * grape
 * almonds
 * olive oil

Vegetable growing (lettuce, artichoke, fennel, cabbage, celery, barattiere, borage, sweet potato, caper, portulaca) and horticulture (kiwis, peaches, orange, clementine, lemon) are also developed.

Viticulture
Vineyards cover 106.715 ha in Apulia, which is 1st place among Italian grape-growing regions. But in the production of quality DOC and DOCG wines, Apulia has only ranked 12th of 20 with 297.667 hl. There are four DOCG wines:
 * Castel del Monte Bombino Nero
 * Castel del Monte Nero di Troia Riserva
 * Castel del Monte Rosso Riserva
 * Primitivo di Manduria Dolce Naturale

Oliviculture
There are an estimated 50 to 60 million olive trees in Apulia, and the region accounts for 40% of Italy's olive oil production. There are four specific Protected designations of Origin (PDO) covering the whole region. Olive varieties include: Baresane, Biancolilla, Brandofino (Castiglione), Buscionetto (Biancolilla), Carolea, Cellina di Nardò, Cerasuola (Ogliara), Cerignola (Bella di Cerignola), Cima di Bitonto, Cima di Mola, Coratina, also grown in Corning, California, a 2018 Gold Medal New York International Olive Oil Competition (NYIOOC) winner, Frantoio, Garganica, La Minuta, Leccino, Moresca, Nocellara Etnea, Nocellara Messinese, Ogliarola, Ogliarola Barese, Ogliara Messinese, Ottobratica, Peranzana, Rotondella, Santagatese, Saracena, Tonda Iblea, and Verdello (subspecies of San Benedetto).

Xylella fastidiosa disease
Since 2008–2010, the olive oil industry in Apulia has been under threat from the pathogen Xylella fastidiosa, a disease that inhibits the trees' uptake of water and nutrients. The epicentre of the epidemic is the southeastern part of the region.

Tourism
Apulia has many small and picturesque villages, 14 of them have been selected by I Borghi più belli d'Italia (The most beautiful Villages of Italy), a non-profit private association of small Italian towns of strong historical and artistic interest, that was founded on the initiative of the Tourism Council of the National Association of Italian Municipalities.

Transport
The region has a good network of roads, but the railway network is less comprehensive, particularly in the south. There are no high-speed lines, but a high-speed line between Naples and Bari is under construction, which should be completed in 2027. The region is crossed northwest to southeast by the A14 highway (Bologna–Taranto), which connects the region's capital, Bari, to Taranto, the second most populous city in the region. The A14 also connects Foggia and points further north along the Adriatic coast to Pescara, Ancona, Rimini and eventually Bologna. The only other highway in the region is the A16 (Naples–Canosa), which crosses the Italian peninsula east–west and links the region with Naples.

There are two international airports, Karol Wojtyła Airport in Bari (IATA: BRI) and Brindisi Airport (IATA: BDS), which serve as the principal logistical hub for the United Nations Global Service Center headquartered in Brindisi. With the approval of a redevelopment project in 2018, the Grottaglie Airfield (IATA: TAR) will host a spaceport for the Italian Space Agency and Virgin Galactic.

Demographics
Emigration from the region's depressed areas to northern Italy and the rest of Europe was very intense in the years between 1956 and 1971. Subsequently, the trend declined, as economic conditions improved, to the point where there was net immigration in the years between 1982 and 1985. Since 1986, the stagnation in employment has led to a new inversion of the trend caused by a decrease in immigration.

Government and politics
Since 1 June 2015, former judge and mayor of Bari Michele Emiliano of the Democratic Party has served as President of the Apulian region.

Administrative divisions

Apulia is divided into six provinces:

Cuisine
Important locally produced ingredients include olive oil, artichokes, tomatoes, eggplant, asparagus, and various kinds of seafood. Local specialties include the carosello, a variety of muskmelon which is often consumed when unripe. Apulian Protected designation of origin (PDO) and Protected Geographical indication (PGI) products included cheeses, olive oils, fruits and vegetables, and a type of bread.

Typically Apulian pasta shapes are orecchiette, cavatelli, and troccoli. Popular street foods include panzerotti, sgagliozze (fried polenta), popizze (small fried balls of pizza dough that are sometimes also called pettule), rustici (puff pastries stuffed with tomatoes, béchamel sauce, mozzarella, and black pepper), the famous pasticiotto (a flaky shortbread dough filled with custard), and focaccia barese. A popular snack in Apulia is Taralli.

Language
As with the other regions of Italy, the national language (since 1861) is Italian. However, because of its long and varied history, other historical languages have been used in this region for centuries. The local languages of northern and central Apulia (roughly the provinces of Bari, Barletta-Andria-Trani, and Foggia as well as the northwestern parts of the Province of Taranto) are the Apulian Southern Italo-Romance dialects, including Bari dialect and Tarantino dialect. In the southern region of Salento, an extreme Southern Italo-Romance language, the Salentino dialect is widely spoken. There is also an Italiot Greek language found in Salento called Griko, which is still spoken by a few thousand Griko people in some areas. In addition, a rare daughter language of the Franco-Provençal language called Faetar is spoken in the mountain villages of Faeto and Celle di San Vito, in the Province of Foggia. It is sometimes classified as a pair of dialects of Franco-Provençal, Faetar and Cellese. The Arbëreshë dialect of the Albanian language has been spoken by a small community since refugees settled there in the 16th century.

Sports
Apulia is home to several national football, water polo, volleyball, basketball and tennis clubs.

Across the top three levels of Italian football, the clubs in Apulia include:
 * U.S. Lecce playing in Serie A
 * S.S.C. Bari playing in Serie B
 * Calcio Foggia 1920 playing in Serie C
 * S.S. Audace Cerignola playing in Serie C
 * Fidelis Andria 2018 playing in Serie C
 * S.S. Monopoli 1966 playing in Serie C
 * Virtus Francavilla Calcio playing in Serie C
 * Taranto F.C. 1927 playing in Serie C

title: Apulia