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Hydraulic ram

A hydraulic ram pump, ram pump, or hydram is a cyclic water pump powered by hydropower. It takes in water at one "hydraulic head" (pressure) and flow rate, and outputs water at a higher hydraulic head and lower flow rate. The device uses the water hammer effect to develop pressure that allows a portion of the input water that powers the pump to be lifted to a point higher than where the water originally started. The hydraulic ram is sometimes used in remote areas, where there is both a source of low-head hydropower and a need for pumping water to a destination higher in elevation than the source. In this situation, the ram is often useful, since it requires no outside source of power other than the kinetic energy of flowing water.

History

The Alhambra, built by Nasrid Sultan Ibn al-Ahmar of Granada beginning in 1238, used a hydram to raise water. Through a first reservoir, filled by a channel from the Darro River, water emptied via a large vertical channel into a second reservoir beneath, creating a whirlpool that in turn propelled water through a much smaller pipe up six metres whilst most water drained into a second, slightly larger pipe.

In 1772, John Whitehurst of Cheshire, England, invented a manually controlled precursor of the hydraulic ram called the "pulsation engine" and installed the first one at Oulton, Cheshire to raise water to a height of 4.9 m. In 1783, he installed another in Ireland. He did not patent it, and details are obscure, but it is known to have had an air vessel.

The first self-acting ram pump was invented by the Frenchman Joseph Michel Montgolfier (best known as a co-inventor of the hot air balloon) in 1796 for raising water in his paper mill at Voiron. His friend Matthew Boulton took out a British patent on his behalf in 1797. The sons of Montgolfier obtained a British patent for an improved version in 1816, and this was acquired, together with Whitehurst's design, in 1820 by Josiah Easton, a Somerset-born engineer who had just moved to London.

Easton's firm, inherited by his son James (1796–1871), grew during the nineteenth century to become one of the more important engineering manufacturers in England, with a large works at Erith, Kent. They specialised in water supply and sewerage systems worldwide, as well as land drainage projects. Eastons had a good business supplying rams for water supply purposes to large country houses, farms, and village communities. Some of their installations still survived as of 2004, one such example being at the hamlet of Toller Whelme, in Dorset. Until about 1958 when the mains water arrived, the hamlet of East Dundry just south of Bristol had three working rams – their noisy "thump" every minute or so resonated through the valley night and day: these rams served farms that needed much water for their dairy herds.

The firm closed in 1909, but the ram business was continued by James R. Easton. In 1929, it was acquired by Green & Carter of Winchester, Hampshire, who were engaged in the manufacturing and installation of Vulcan and Vacher Rams.

The first US patent was issued to Joseph Cerneau (or Curneau) and Stephen (Étienne) S. Hallet (1755-1825) in 1809. US interest in hydraulic rams picked up around 1840, as further patents were issued and domestic companies started offering rams for sale. Toward the end of the 19th century, interest waned as electricity and electric pumps became widely available.

Priestly's Hydraulic Ram, built in 1890 in Idaho, was a "marvelous" invention, apparently independent, which lifted water 110 ft to provide irrigation. The ram survives and is listed on the U.S. National Register of Historic Places.

By the end of the twentieth century, interest in hydraulic rams has revived, due to the needs of sustainable technology in developing countries, and energy conservation in developed ones. An example is Aid Foundation International in the Philippines, who won an Ashden Award for their work developing ram pumps that could be easily maintained for use in remote villages. The hydraulic ram principle has been used in some proposals for exploiting wave power, one of which was discussed as long ago as 1931 by Hanns Günther in his book In hundert Jahren.

Some later ram designs in the UK called compound rams were designed to pump treated water using an untreated drive water source, which overcomes some of the problems of having drinking water sourced from an open stream.

In 1996 English engineer Frederick Philip Selwyn patented a more compact hydraulic ram pump where the waste valve used the venturi effect and was arranged concentrically around the input pipe. Initially patented as a fluid pressure amplifier due to its different design, it is currently sold as the "Papa Pump". Additionally to this a large scale version named the "Venturo Pump" is also being manufactured.

Construction and principle of operation
A traditional hydraulic ram has only two moving parts, a spring or weight loaded "waste" valve sometimes known as the "clack" valve and a "delivery" check valve, making it cheap to build, easy to maintain, and very reliable.

Priestly's Hydraulic Ram, described in detail in the 1947 Encyclopedia Britannica, has no moving parts.

Sequence of operation
A simplified hydraulic ram is shown in Figure 2. Initially, the waste valve [4] is open (i.e. lowered) because of its own weight, and the delivery valve [5] is closed under the pressure caused by the water column from the outlet [3]. The water in the inlet pipe [1] starts to flow under the force of gravity and picks up speed and kinetic energy until the increasing drag force lifts the waste valve's weight and closes it. The momentum of the water flow in the inlet pipe against the now closed waste valve causes a water hammer that raises the pressure in the pump beyond the pressure caused by the water column pressing down from the outlet. This pressure differential now opens the delivery valve [5], and forces some water to flow into the delivery pipe [3]. Because this water is being forced uphill through the delivery pipe farther than it is falling downhill from the source, the flow slows; when the flow reverses, the delivery check valve [5] closes. Meanwhile, the water hammer from the closing of the waste valve also produces a pressure pulse which propagates back up the inlet pipe to the source where it converts to a suction pulse that propagates back down the inlet pipe. This suction pulse, with the weight or spring on the valve, pulls the waste valve back open and allows the process to begin again. A pressure vessel [6] containing air cushions the hydraulic pressure shock when the waste valve closes, and it also improves the pumping efficiency by allowing a more constant flow through the delivery pipe. Although the pump could in theory work without it, the efficiency would drop drastically and the pump would be subject to extraordinary stresses that could shorten its life considerably. One problem is that the pressurized air will gradually dissolve into the water until none remains. One solution to this problem is to have the air separated from the water by an elastic diaphragm (similar to an expansion tank); however, this solution can be problematic in developing countries where replacements are difficult to procure. Another solution is a snifting valve installed close to the drive side of the delivery valve. This automatically inhales a small amount of air each time the delivery valve shuts and the partial vacuum develops. Another solution is to insert an inner tube of a car or bicycle tire into the pressure vessel with some air in it and the valve closed. This tube is in effect the same as the diaphragm, but it is implemented with more widely available materials. The air in the tube cushions the shock of the water the same as the air in other configurations does.

Efficiency
A typical energy efficiency is 60%, but up to 80% is possible. This should not be confused with the volumetric efficiency, which relates the volume of water delivered to total water taken from the source. The portion of water available at the delivery pipe will be reduced by the ratio of the delivery head to the supply head. Thus if the source is 2 m above the ram and the water is lifted to 10 m above the ram, only 20% of the supplied water can be available, the other 80% being spilled via the waste valve. These ratios assume 100% energy efficiency. Actual water delivered will be further reduced by the energy efficiency factor. In the above example, if the energy efficiency is 70%, the water delivered will be 70% of 20%, i.e. 14%. Assuming a 2-to-1 supply-head-to-delivery-head ratio and 70% efficiency, the delivered water would be 70% of 50%, i.e. 35%. Very high ratios of delivery to supply head usually result in lowered energy efficiency. Suppliers of rams often provide tables giving expected volume ratios based on actual tests.

Drive and delivery pipe design
Since both efficiency and reliable cycling depend on water hammer effects, the drive pipe design is important. It should be between 3 and 7 times longer than the vertical distance between the source and the ram. Commercial rams may have an input fitting designed to accommodate this optimum slope. The diameter of the supply pipe would normally match the diameter of  the input fitting on the ram, which in turn is based on its pumping capacity. The drive pipe should be of constant diameter and material, and should be as straight as possible. Where bends are necessary, they should be smooth, large diameter curves. Even a large spiral is allowed, but elbows are to be avoided. PVC will work in some installations, but steel pipe is preferred, although much more expensive. If valves are used they should be a free flow type such as a ball valve or gate valve.

The delivery pipe is much less critical since the pressure vessel prevents water hammer effects from traveling up it. Its overall design would be determined by the allowable pressure drop based on the expected flow. Typically the pipe size will be about half that of the supply pipe, but for very long runs a larger size may be indicated. PVC pipe and any necessary valves are not a problem.

Starting operation
A ram newly placed into operation or which has stopped cycling should start automatically if the waste valve weight or spring pressure is adjusted correctly, but it can be restarted as follows: If the waste valve is in the raised (closed) position, it must be pushed down manually into the open position and released. If the flow is sufficient, it will then cycle at least once. If it does not continue to cycle, it must be pushed down repeatedly until it cycles continuously on its own, usually after three or four manual cycles. If the ram stops with the waste valve in the down (open) position it must be lifted manually and kept up for as long as necessary for the supply pipe to fill with water and for any air bubbles to travel up the pipe to the source. This may take some time, depending on supply pipe length and diameter. Then it can be started manually by pushing it down a few times as described above. Having a valve on the delivery pipe at the ram makes starting easier. Closing the valve until the ram starts cycling, then gradually opening it to fill the delivery pipe. If opened too quickly it will stop the cycle. Once the delivery pipe is full the valve can be left open.

Common operational problems
Failure to deliver sufficient water may be due to improper adjustment of the waste valve, having too little air in the pressure vessel, or simply attempting to raise the water higher than the level of which the ram is capable.

The ram may be damaged by freezing in winter, or loss of air in the pressure vessel leading to excess stress on the ram parts. These failures will require welding or other repair methods and perhaps parts replacement.

It is not uncommon for an operating ram to require occasional restarts. The cycling may stop due to poor adjustment of the waste valve, or insufficient water flow at the source. Air can enter if the supply water level is not at least a few inches above the input end of the supply pipe. Other problems are blockage of the valves with debris, or improper installation, such as using a supply pipe of non-uniform diameter or material, having sharp bends or a rough interior, or one that is too long or short for the drop, or is made of an insufficiently rigid material. A PVC supply pipe will work in some installations but a steel pipe is better.

title: Hydraulic ram


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Talk:Homomorphism

Identity is not always preserved
Consider the map f:(Z,.)--->(Z,.) (Z: set of integers, . : multiplication) between monoids given by f(x)=0. This is a homomorphism since f(x.y) = f(x).f(y) but identity is not preserved since f(1) = 0. The statement that identity is preservd seems wrong to me. --Shahab 12:14, 1 September 2006 (UTC)

 * Doesn't a homomorphism need to preserve identity? The definition of monoid homomorphisms in the article on monoids requires this; the same goes for the definition of functors (and it says in the article about functors that one can think of them as homomorphisms between categories). 95.88.237.20 (talk) 17:36, 12 February 2013 (UTC)
 * In any variety (or a more general class) of algebras, homomorphisms are required to preserve all operations in the signature of the variety. The signature of monoids includes the constant for identity, hence monoid homomorphisms have to preserve identity elements. The signature of, say, semigroups does not include identity, hence semigroup homomorphisms do not have to preserve identity (even if the two semigroups happen to be monoids). Shahab’s map is a semigroup homomorphism, but not a monoid homomorphism.—Emil J. 17:55, 12 February 2013 (UTC)

 * Indeed, this is quite a nice illustration of the necessity for care over the codomain as well as the domain of a map. The map x → 0 is a monoid homomorphism from Z,× to the monoid {0},× but not to the monoid Z,×.  Of course, {0},× is a monoid, and a subsemigroup of Z,×, but not a submonoid of Z,× since the identity of {0},× is not the identity of Z,×.  Deltahedron (talk) 20:33, 12 February 2013 (UTC)

 * The wording under the Definition section is confusing in this respect. It suggests on first reading, for example, that a group homomorphism only applies between two groups, and not between say two rings (though on further thinking, of course every ring is a group under addition). EmilJ and Deltahedron make it clear here; we could do with this clarity in the article. I'll try to reword the definition to make it clearer that the structure that is preserved by a homomorphism is nominated (and only implied as a default), and not specified by the objects. In particular, it should be clear that a "monoid homomorphism" is a homomorphism that preserves all structure that is required by the definition of monoids. My attempt may break subtler aspects, please feel free to rescue the correction. — Quondum (talk) 08:31, 24 September 2013 (UTC)

Poorly Explained
This page should follow the example of the Group page and have formal definitions as well as examples because definitions may be helpful, but understanding won't come with a plethora of unorganized information. Therazzz (talk) 22:20, 23 May 2014 (UTC)

Isomorphism if and only if both a monomorphism and an epimorphism
The statement that "in abstract algebra a homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism" needs qualification. The definition in terms of injective and surjective maps makes this true, but the categorical definition in terms of left and right cancellation does not. For example, the injection i : Z → Q is a ring monomorphism and epimorphism in the latter sense, but not an isomorphism. See, for example,. Deltahedron (talk) 18:14, 11 July 2014 (UTC)
 * I agree. I have tagged the disputed definitions. It seems, from the comments in the text that previously these definitions were explicitly restricted to modules and that the present version is due to a troll that believed that abstract algebra is reduced to module theory. I have not yet got the time to check the history. D.Lazard (talk) 19:49, 11 July 2014 (UTC)
 * In fact, the wrong definitions have been introduced by this edit . Should we restore the previous version of the section or rewrite the section to have a correct version? D.Lazard (talk) 22:10, 11 July 2014 (UTC)

Sorry, but I don't understand your point. First, a definition can't be wrong, but unusual at worst. Second, the case of category theory is handled in an own subsection where the inclusion Z-->Q is even used as example. The text there appears to say exactly the same as Deltahedron. So, where is the problem? - Jochen Burghardt (talk) 22:28, 11 July 2014 (UTC) Btw: It was my edit, and I don't think I deserve to be called a troll. - Jochen Burghardt (talk) 22:32, 11 July 2014 (UTC)

 * Your edit assert implicitly that "monomorphism", "epimorphism", ... have two non-equivalent meanings in mathematics. This is this implicit assertion which is wrong. Saying, as you did that in abstract algebra an epimorphism is always surjective, implies that this is true for rings or that the study of rings do not belong to abstract algebra; both are wrong. I remove the word of troll. Your edit is only original research that is highly confusing for the reader and has been inserted with a summary saying that the previous correct version was confusing. D.Lazard (talk) 23:19, 11 July 2014 (UTC)

Please have a look at the version immediately before my edit. It says:  "... However, the definitions in category theory are somewhat technical. In the important special case of module homomorphisms, and for some other classes of homomorphisms, there are much simpler descriptions, as follows: ... An epimorphism (sometimes called a cover) is a surjective homomorphism. ... For endomorphisms and automorphisms, the descriptions above coincide with the category theoretic definitions; the first three descriptions do not. ..."  So the non-equivalent meanings of (e.g.) epimorphism were present already before my edit, as was the definition of epimorphism to be a surjective homomorphism (which happens to be the definition I am familar with).

If you look at Epimorphism, it gives the right-cancellation definition in the context of category theory, and notes a deviating definition (viz. being surjective) by "Many authors in abstract algebra and universal algebra". This is consistent with the result of my edit.

You are right that we should supply a citation for the non-category wording. I can offer the following one: Garrett Birkhoff, "Lattice Theory", Providence, Am. Math. Soc., Vol.25 of Colloquium Publications, 1967, Sect.VI.3, p.134 (although the overall issue of the book are lattices, Sect.VI is on "Universal Algebra", and VI.3 is on "Morphisms"). Birkhoff says:  A morphism of A onto B is called an epimorphism; a one-one morphism is called a monomorphism. An isomorphism of A with B is defined to be a one-one morphism from A onto B; an automorphism is an isomorphism of an algebra with itself, and an endomorphism is a morphism of an algebra into itself.  - Jochen Burghardt (talk) 07:09, 12 July 2014 (UTC)

Inaccuracy (remainder)
This has already been pointed out in 2008 at Talk:Homomorphism/Archive 1 but, as a remainder, I just want to point out the definitions of an isomorphism and an epimorphisms given in the article are ... wrong. The simplest way to see it to look them at some algebra textbooks. But more conceptually the definitions should not refer to the underlying set; this allows in particular a smooth transition from sets to some more structured objects; for instance, one can talk about a homomorphism between sheaves of modules. -- Taku (talk) 20:28, 30 November 2016 (UTC)
 * I agree. Moreover, the "equivalently" assertion for epimorphisms is wrong, even for modules. I'll try to correct the section, if you do not do it before me. D.Lazard (talk) 09:20, 1 December 2016 (UTC)

 * Being not a native English speaker, the only relevant textbook I have at hand is Birkhoff.1967, cited above, where and I had almost the same diccsussion in 2014. As a result, I then tried to fix section Homomorphism according to Birkhoff's definitions. If they are unusual today, we need other citations supporting that - nobody gave some since 2014.

 * In any case, I suggest to keep the remarks (and also the proofs) on relations between the category theory and the (i.e. Birkhoff's) abstract algebra notions; this is possible even if "iso", "epi", "mono" is used only for the former, and only "bijective", "onto", "one-one" for the latter. Moreover, the article's current version uses terms like "generally", "often" etc. in places where definitions are expected, and we should avoid that.

 * If there isn't sufficient evidence that Birkhoff's terminology isn't used any longer today, I suggest just to revert D.Lazard's edits. The ring homomorphism example was already present in the 15-Jul-2016 version, and the module homomorphism example can easily be added to the same paragraph; a more explicit warning about deviating notions in abtract algebra and category theory then should be added there, too. - Jochen Burghardt (talk) 13:43, 6 December 2016 (UTC)
 * In your above citation, it is not said what are A and B. The sentence is undoubtful correct if they are sets, modules or groups. If Birkhoff allows more general categories for them, this definition is clearly in contradiction with the present standard of mathematics. One must keep in mind that when he wrote his book, the language of categories was not yet popularized in all mathematics. Therefore, this book is not reliable for the modern terminology. Moreover, for some authors the term "homomorphism" is used only for algebraic structures, while, for others, this is a synonymous of "morphism". As we cannot know the background of the readers, we must have a formulation which is correct, whichever is this background. This is the reason for giving definitions that are correct in any case, and then explain that "in general" isomorphism = bijective, monomorphism = injective and epimorphism = surjective, with a short explanation of what "in general" means in each case. This is what I have tried to do with my edit. In any case, I am strongly against reverting to previous version, as 1/ it may be confusing for people who do not care that the given definition do not apply for rings and non-algebraic structures. 2/ it contains the wrong assertions that every epimorphism is a split epimorphism and that every monomorphism is a split monomorphism (this is true for vector spaces, but not for groups an modules). D.Lazard (talk) 15:32, 6 December 2016 (UTC)
 * So I understand you havn't an Englush textbook available either? Possibly, (Taku) can help with a few citations of algebra textbooks? - Jochen Burghardt (talk) 18:59, 6 December 2016 (UTC)
 * This is not a problem of language nor a problem of availability of textbooks. The point is that this article is about a general definition of homomorphism, and that most definitions given in textbooks are about specific kinds of homomorphisms, commonly, homomorphisms of groups, vector spaces and modules. As all definitions are equivalent in these cases, these textbooks are not useful for sourcing a general definition. As far as I know, there exist three theories providing a general definition of isomorphism, epimorphism and monomorphism. The first one is Bourbaki's definition of algebraic structures. It has never been accepted by the mathematicians community, nor really used by other authors, and thus cannot be used as a source. The second one is the theory of universal algebra. Although being the object of some active research, universal algebra has rarely been considered in the main stream of mathematics, and its terminology is not widely accepted; a witness of this is that the term "variety" has a completely different meaning in universal algebra and in the main stream of mathematics (algebraic variety). Thus universal algebra, and Birkhoff book may not be used as a source for the terminology of this article. The last general definition of homorphisms that I know is that of category theory. It is presently used in many fields of mathematics, from number theory (Wiles's proof of Fermat's Last Theorem) to topology (via homological algebra). It follows that the only terminology that make sense for this article is that of category theory. Any other terminology would be confusing for many readers. However, I agree that if some sources use a terminology which is not compatible, this must be added to the article in a sentence like "Some author use the following different definition ...". D.Lazard (talk) 10:55, 7 December 2016 (UTC)

 * Ok, now I understand your point of view. What about the following suggestion:
 * We use adjectives to name the (homo)morphism properties, as far as possible, i.e. "monic", "epic" (this is what S. Mac Lane uses in the cited book, p.19), "injective", "surjective", as there is no confusion about them. We discuss the differences and relations between them (such as "surjective implies epic, but nor vice versa: e.g. ring homomorphism Z-->Q"). We comment on different terminologies and warn about confusions, using the nouns ("monomorphism", "epimorphism") for that. We could e.g. write "In category theory and some abstract algebra books, "monomorphism" and "epimorphism" refers to a monic and epic morphism, respectively,(some citations) while other abstract algebra books use these notions to refer to an injective and surjective homomorphism, respectively.Birkhoff.1967, Stanley.Sankappanavar.2012:29,43" We could extend the former image to include all kinds of morphisms discussed, by adding "inj" and "surj" and extending the areas for "mon" and "epi" (then meaning "monic" and "epic") appropriately. And of course we should warn that "injective" and "surjective" don't make sense in category theory (if I remember right - ?).
 * Our overall goal should be that a by reading the section under consideration, the confusion (that caused our repeated discussions here) can be overcome. - Jochen Burghardt (talk) 13:45, 7 December 2016 (UTC)

 * I agree with your overall goal, but not with the suggested use of epic and monic. In fact, making a difference between "being an epimorphism" and "being epic" is WP:OR, and cannot be introduced in Wikipedia. IMO, the problem with the present state of section "Types" is that the bulleted list implies a very short description, and that this description, although correct (in my opinion) it too short for being clearly understood by a non-expert reader. Therefore, I suggest to merge the subsection in the section and to split the section in subsections, one for each item of the bulleted list. This would allow to clarify everything, to provide as many examples as needed, and so on. D.Lazard (talk) 15:16, 7 December 2016 (UTC)

 * No principal objection, but I see a problem of presentation: in each subsection we'd repeat something like "some authors call it '...', and some call it '...'", and the information was lost that the same authors that call an injective homomorphism "monomorphism" also use "epimorphism" for a surjective one. It is easier to present that information if we first split into subsections like e.g. "category theory approaches" and e.g. "set theory approaches". The suffix "approaches" is intended to reflect that the notions inspired by category theory are nowadays used beyond that field, in particular in abstract algebra (this was Taku's and your point above, if I understand you right); maybe you have better title suggestions to express this intention. - Jochen Burghardt (talk) 20:12, 7 December 2016 (UTC)

The presentation of this materials (epimorphisms, etc) is still not optimal. I have come to wonder: do we really need to give definitions of epimorphisms and such for some hypothetical algebraic object? Since, as I want to stress, an algebraic object need not have an underlying set, the set-theoretic definition cannot be used for that purpose (this was a point I wanted to make). It seems a solution among contemporary textbook authors (Dummit-Foote, Eisenbud) is simply to avoid such problematic terms like an epimorphism and use more down-to-earth unambiguous terms like surjective homomorphisms or injective morphisms. They then discuss category theory and mentions, for instance, a subjective module homomorphism is precisely an epimorphism in the category of modules. Put in another way, do people use terms like epimorphisms aside from the category-theoretic meaning? Yes, some do but that's limited to historical sources. (There is no neee to use historical terms in Wikipedia.) -- Taku (talk) 03:50, 11 December 2016 (UTC)

 * I think it is a good idea to use unambiguous names in definitions, and that "surjective", "injective" (from your citations), and "left" and "right cancellable" (from Mac Lane's "Categories for the Working Mathematician") is a good choice. Based on these definitions, we could then say that "epimorphism" and "monomorphism" in earlier days meant "surjective" and "injective" homomorphism ("set-theoretically inspired meaning"), but has got a second ("category-theoretically inspired") meaning with the raise of category theory, viz. "left" and "right cancellable" (homo)morphism, respectively. If I understand and you right, it is your point that the latter meaning is more elegant and therefore gets more and more widespread in abstract algebra, even beyond pure category theory; this should be made clear in the article (provided there is evidence for it).
 * Concerning your second question, please note that up to now the article only has citations that support the set-theoretic meanings (viz. Birkhoff.1967, which is admittedly historical, and Burris.Sankappanavar.2012, which is undoubtedly not), apart from the pure category-theory book Saunders.1971 (which moreover isn't much more contemporary than Birkhoff.1967). So, if you can provide citations that support your point, please do! In particular, could you please give full references of the "Dummit-Foote, Eisenbud" textbook(s)? Since the set-theory meaning is still (widely, I believe, based on the current distribution of citations) in use, the article should mention it; from your point of view, readers should be warned of the unelegent meaning still found in the literature. - Jochen Burghardt (talk) 09:56, 11 December 2016 (UTC)

 * I agree with your suggestion of using "surjective", "injective", "left" and "right cancelable". However, we have also to discuss all the names, their variants, and their relationships. As this discussion cannot be done in a single line, this is the main reason for splitting the section into one subsection by concept.
 * IMO, most of this discussion results of an ambiguity in the first line of the article: the definition of "algebraic structure", given in the linked article, is not recalled here. Thus it is unclear that, here, an algebraic structure has a underlying set, and only one, while, for many mathematicians, "algebraic structure" is either not clearly defined or is much larger concept. Thus, we have first to state clearly that in this article, an algebraic structure has one and only one underlying set.
 * About "isomorphisms": It is a (rather easy) theorem that, for these algebraic structures, bijective is equivalent with having an inverse. This must be presented as a theorem, and possibly proved.
 * About "monomorphisms": It is also a theorem that, for algebraic structures for which all operations are everywhere defined (varieties of universal algebra), "monomorphism" is equivalent with "injective". I do not know of an explicit source, but the proof may proceed as follows. Given a variety of algebraic structure, a free object over x is an algebraic structure of the variety that has a distinguished element x and has the property that for any element a of any algebraic structure of the variety, there exists an homomorphism from the free object to the algebraic structure that maps x to a. The existence of a free object over x implies, almost immediately that every monomorphism is injective, and thus that "monomorphism" is equivalent with "injective". The free object over x, which is unique up to an isomorphism is, for sets, any singleton, for semigroups, the semigroup of positive integers, for monoids, the monoid of nonnegative integers, for groups, the additive group of the integers, for rings, the polynomial ring in x with integer coefficients, and for rngs, the subset of the preceding consisting of the polynomials without constant terms. For vector spaces and modules, the free object over x is the vector space or free module which has {x} as a base. In general, the free object over x may be constructed by considering the formal application of the operations of the structure to x. This cannot be done if some operations are not everywhere defined, such as the multiplicative inverse in the case of field.
 * It would be useful to find an explicit reference for this result (it is certainly not new). If we do not find such a reference, I suggest to include the proof in the article, because this would avoid the vague "in may cases" D.Lazard (talk) 14:10, 11 December 2016 (UTC)

 * (Sorry, by "Dummit-Foote, I meant their "abstract algebra" and by "Eisenbud", "communicative algebra view toward algebraic geometry". I have deliberately avoided old, foreign or other less popular textbooks. The fact they avoid the term "epimorphism" is a good indication that we should do the same in Wikipedia.)
 * I think the problem of the confusing state of the sectio stems from our attempt to give general definitions of isomorphism, epimorphism, etc. for general algebraic structures. Since the latter is not well-defined, that's not possible and we shouldn't try that. An "epimorphism" has specific meaning in the category theory but some authors, especially in historical sources, use it also mean a surjective homomorphism. We should stick to such a commentary on the terminology and not try to give a mathematical exposition.
 * As for an isomorphism, before defining it, if we like, we can explain it along the line: if a homomorphism is a bijection between the underlying sets (note algebraic structure is not a set!!), then the inverse function is necessarily a homomorphism. And, as D.Lazard suggested, with a proof, since the proof is illuminating. --- Taku (talk) 21:54, 12 December 2016 (UTC)

 * Could you please give a full citation of your books? Apparently, they are the only references up to now that could support your and 's claim that category theory notions are commonly used in abstract algebra, so we badly need them. I've been searching in our library, but found only these books, all sticking to set-based notions:
 * — defines injection, surjection (p.3); unsorted term algebra, unsorted homomorphism, endomorphism (p.17); many-sorted algebra, many-sorted homomorphism (p.246) — the latter may be a good source for algebras with more than one carrier set
 * — defines algebra (p.387, single carrier-set only), homomorphsim (p.395, for algebras with operations of an arity up to 2 only), surjective=onto function and injective=one-to-one function in general (p.282)
 * — doesn't handle abstract algebras (although he mentions them informally on p.56,59); defines particular algebras, like group (p.258) and group homomorphism (p.267); defines injective=one-to-one=monic function and surjective=onto=epic function (p.161) in general
 * In order to ease discussion, I suggest to upload scans of the most relevant pages locally for a short time to this page - I guess this could be allowed by the fair use rule. Or does somebody see a problem with that?
 * I agree to include some elementary proofs into the article; in the version of 26 Nov 2016, there were already three proofs of that kind. If they are re-inserted, they should get a more explicit header such that it becomes obvious what they are about without having to read the corresponding section in the article. (I guess, due to the latter problem, removed them on 1 Dec.) - Jochen Burghardt (talk) 13:14, 3 January 2017 (UTC)
 * It is not because I am against proofs in WP that I have removed these three proofs. It because they were confusing, as the statements that were supposed to be proved were lacking.
 * By the way, I have started to write a new version for the section "Types" (see User:D.Lazard/sandbox). Presently, only the end of the (collapsed) proof for monomorphism, and the subsection about epimorphisms are is sill lacking. Here are some comments about this rewriting.
 * I have tried to avoid the above debate in stating that the two definitions of monomorphism are essentially equivalent. As I do not know a good reference for this equivalence, I have included a proof, collapsed, because it is too technical and not really interesting for most readers.
 * For epimorphisms, I do not know a condition implying the equivalence, nor the details of how to write the subsection.
 * I have encountered a problem of terminology: firstly it is not clear, what exactly means "algebraic structure", a specific instance, as the additive group of the integers or the whole class, as "structure of group". In general, and in Algebraic structure, an algebraic structure is a triple consisting of a set, the operations and the axioms satisfied by these operations. But it seems that there is no commonly accepted term for the class of algebraic structures sharing the same operations and the same axioms. Bourbaki uses "species of algebraic structure", but it seems that this terminology is generally ignored. In universal algebra, one use the term of variety, but the term is too restricted, as fields do not form a variety. Moreover most people interested in algebra do not know the terminology of universal algebra. Thus, for not being too technical, some caution is needed for using the terminology of universal algebra in this article. Because of this, I am not fully satisfied by some of my sentences, and some help would be welcome.
 * I intend to replace section "Types" by my new version. Before that, it would be better to have a consensus that this replacement will improve the article. D.Lazard (talk) 15:32, 3 January 2017 (UTC)

New section "Special homomorphisms"
I have completely rewritten, and expanded the section "Types", now renamed "Special homomorphisms". In my opinion, this solves the questions discussed in the previous section. Nevertheless some further work would be useful, in particular for improving references and settling the points that I have listed at the end of the preceding section. D.Lazard (talk) 14:26, 6 January 2017 (UTC)

Figure
Some visitors might be looking for a quick understanding of these concepts in the common case of sets, groups, etc. For beginners especially, would including this figure be helpful? Book here. Also might address the removed figure mentioned above. Adam Marsh (talk) 23:53, 26 January 2018 (UTC)

 * This figure is not convenient for this article, because it is wrong in this context. In fact, a large part of this article is devoted to showing that monomorphism and epimorphism are not synonymous of injective and surjective (respectively). Moreover, this figure introduces a confusion between "function" and "homomorphism": looking at the figure, a beginner could think that the hyperbolic sine is an automorphism of the reals, which is blatantly wrong. D.Lazard (talk) 10:38, 27 January 2018 (UTC)

 * Ok. I suppose an implicit question I had was whether the category theoretic definitions might be better placed either under a separate subheading or under morphism as opposed to here. I would guess that at least some readers landing here might be confused by the introduction of the more general definitions, and could be looking for the algebraic definitions (injective / surjective). Not sure I understand the comment about functions, the assumption was that the figure would be placed in context, which is clearly that of homomorphisms, not functions. Adam Marsh (talk) 04:00, 1 February 2018 (UTC)

Epimorphisms of groups
In January 2017, I edited the article for clarifying the relationships monomorphism–injection and epimorphism–surjection. Reading my edits again, I remark that I have left open the question of (non-necessarily commutative) groups: is every epimorphism (categorical meaning) of groups a surjection? More precisely, do there exist a right-cancelable group homomorphism that is not surjective? Does anybody know an answer to this question? D.Lazard (talk) 10:31, 7 May 2018 (UTC)
 * I have found a reference, and I'll edit the article accordingly. D.Lazard (talk) 13:14, 7 May 2018 (UTC)

All algebraic structures are homomorphic
Should we include the following in the article?

Let $$(R, +_R, \times_R)$$ and $$(S, +_S, \times_S)$$ be rings and let $$\phi: R \to S$$ be the trivial homomorphism, i.e. the homomorphism mapping every element of $$R$$ to the additive identity $$0_S$$ of $$S$$. This is indeed a homomorphism; we have for all $$r, r' \in R$$, $$\phi(r +_R r') = \phi(r) +_S \phi(r')$$ and $$\phi(r \times_R r') = \phi(r) \times_S \phi(r')$$ since both sides of both equations evaluate to $$0_S$$. Indeed this works not only with rings or groups, but with any algebraic structure; you can add as many operations as you want, binary or otherwise, and the argument still works. Therefore all groups are homomorphic, as are all rings, and indeed all algebraic structures. This is why we have a notation $$\cong$$ for isomorphy but no notation for homomorphy. Joel Brennan (talk) 20:01, 5 July 2018 (UTC)
 * Being homomorphic is not a relevant concept, as, for most structures, there are many homomorphisms between any two objects. Your example is not a ring homomorphism, as $$\phi(1_R)\ne 1_S.$$ There is no ring homomorphism between two fields of different characteristic. Your example works only for structures having a zero object. Nevertheless a few words may be added to the article for saying that many algebraic structures have zero (homo)morphisms.
 * Thank you for correcting my example; I was not aware that preservation of the multiplicative identity was required in the definition of ring homomorphism, and as it turns out neither was the lecturer for my rings course at univeristy, whom I have now informed. However does this not then mean that we also should alter the definition of a general homomorphism in this article; the general definition given here does not line up with the definition of ring homomorphism given in its corresponding article. Unless of course the definition given on that wikipedia page is really of a non-zero homomorphism, which should be clarified over there. Joel Brennan (talk) 20:36, 7 July 2018 (UTC)
 * The definition that is given here must not be changed. However, it is useful to clarify that is applies also to 0-ary operations. I have added a paragraph in the article for this clarification. D.Lazard (talk) 07:26, 8 July 2018 (UTC)
 * Thank you that is a perfect explanation. Joel Brennan (talk) 11:07, 8 July 2018 (UTC)

Section on the history and application of the homomorphism concept?
I think it would be good if we could add a section on the history of the homomorphism concept. In particular: Who invented the concept? Which motivation did he have? For which specific purposes is the concept used today? Zaunlen (talk) 23:48, 7 November 2019 (UTC)

Operation version?
What is operation version? The sentence is: "Formally, a map $f: A\to B$ preserves an operation $\mu$ of arity k, defined on both $A$ and $B$ if
 * $f(\mu_A(a_1, \ldots, a_k)) = \mu_B(f(a_1), \ldots, f(a_k)),$"

This sentence says that "preserves ««an»» operation". «an» means "a single" or "untyped" or "unversioned". If we should use versions, the sentence should be modified and make it "preserves operations $\mu_A and \mu_B$ (versions of $\mu$) of arity k,"

There is something wrong here: we should either change the sentence, or change the formula, or somehow clarify the meaning of subscripts in one or two sentences. Hooman Mallahzadeh (talk) 16:47, 31 May 2022 (UTC)
 * The fact that $A$ and $B$ have the same type, means that there is a one to one correspondence between their operations, and that the corresponding operations have the same arity and satisfy the same axioms. In most cases, the corresponding operations have the same name, but this is not required. An example is the exponential function, which is an homomorphism from the additive group of the complex numbers to the multiplicative group of the nonzero complex numbers. So, in the definition, one must distinguish between an operation on $A$, and the corresponding operation on $B$. The easiest way for that is to index the name of the operation with the name of the structure.
 * I agree that presently, the formulation is confusing, but introducing the concept of "operation version" does not solve anything. Instead, the beginning of the section shoud be rewritten in the line of what precedes. D.Lazard (talk) 17:16, 31 May 2022 (UTC)

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

title: Talk:Homomorphism


text:
Homininae

Homininae, also called "African hominids" or "African apes", is a subfamily of Hominidae. It includes two tribes, with their extant as well as extinct species: 1) the tribe Hominini (with the genus Homo including modern humans and numerous extinct species; the subtribe Hominina, comprising at least two extinct genera; and the subtribe Panina, represented only by the genus Pan, which includes chimpanzees and bonobos)―and 2) the tribe Gorillini (gorillas). Alternatively, the genus Pan is sometimes considered to belong to its own third tribe, Panini. Homininae comprises all hominids that arose after orangutans (subfamily Ponginae) split from the line of great apes. The Homininae cladogram has three main branches, which lead to gorillas (via the tribe Gorillini) and to humans and chimpanzees (via the tribe Hominini and subtribes Hominina and Panina―see the evolutionary tree below). There are two living species of Panina (chimpanzees and bonobos) and two living species of gorillas, but only one extant human species. Traces of extinct Homo species, including Homo floresiensis have been found with dates as recent as 40,000 years ago. Organisms in this subfamily are described as hominine or hominines (not to be confused with the terms hominins or Hominini).

History of discoveries and classification
Until 1970, the family (and term) Hominidae meant humans only; the non-human great apes were assigned to the family Pongidae. Later discoveries led to revised classifications, with the great apes then united with humans (now in subfamily Homininae) as members of family Hominidae By 1990, it was recognized that gorillas and chimpanzees are more closely related to humans than they are to orangutans, leading to their (gorillas' and chimpanzees') placement in subfamily Homininae as well.

The subfamily Homininae can be further subdivided into three branches: the tribe Gorillini (gorillas), the tribe Hominini with subtribes Panina (chimpanzees) and Hominina (humans and their extinct relatives), and the extinct tribe Dryopithecini. The Late Miocene fossil Nakalipithecus nakayamai, described in 2007, is a basal member of this clade, as is, perhaps, its contemporary Ouranopithecus; that is, they are not assignable to any of the three extant branches. Their existence suggests that the Homininae tribes diverged not earlier than about 8 million years ago (see Human evolutionary genetics).

Today, chimpanzees and gorillas live in tropical forests with acid soils that rarely preserve fossils. Although no fossil gorillas have been reported, four chimpanzee teeth about 500,000 years old have been discovered in the East-African rift valley (Kapthurin Formation, Kenya), where many fossils from the human lineage (hominins) have been found. This shows that some chimpanzees lived close to Homo (H. erectus or H. rhodesiensis) at the time; the same is likely true for gorillas.

Taxonomic classification
Homininae
 * Tribe Dryopithecini†
 * Kenyapithecus (?)
 * Kenyapitheus wickeri
 * Ouranopithecus
 * Ouranopithecus macedoniensis
 * Otavipithecus
 * Otavipithecus namibiensis
 * Oreopithecus (?)
 * Oreopithecus bambolii
 * Nakalipithecus
 * Nakalipithecus nakayamai
 * Anoiapithecus
 * Anoiapithecus brevirostris
 * Dryopithecus
 * Dryopithecus fontani
 * Hispanopithecus (?)
 * Hispanopithecus laietanus
 * Hispanopithecus crusafonti
 * Pierolapithecus
 * Pierolapithecus catalaunicus
 * Rudapithecus (?)
 * Rudapithecus hungaricus
 * Samburupithecus
 * Samburupithecus kiptalami
 * Danuvius
 * Danuvius guggenmosi
 * Tribe Gorillini
 * Chororapithecus †
 * Chororapithecus abyssinicus
 * Genus Gorilla
 * Western gorilla, Gorilla gorilla
 * Western lowland gorilla, Gorilla gorilla gorilla
 * Cross River gorilla, Gorilla gorilla diehli
 * Eastern gorilla, Gorilla beringei
 * Mountain gorilla, Gorilla beringei beringei
 * Eastern lowland gorilla, Gorilla beringei graueri
 * Tribe Hominini
 * Subtribe Panina
 * Genus Pan
 * Chimpanzee (common chimpanzee), Pan troglodytes
 * Central chimpanzee, Pan troglodytes troglodytes
 * Western chimpanzee, Pan troglodytes verus
 * Nigeria-Cameroon chimpanzee, Pan troglodytes ellioti
 * Eastern chimpanzee, Pan troglodytes schweinfurthii
 * Bonobo (pygmy chimpanzee), Pan paniscus
 * Subtribe Hominina
 * Graecopithecus †
 * Graecopithecus freybergi. Note: Graecopithecus has also been subsumed by other authors into Dryopithecus. The placement of Graecopithecus within the Hominina, as shown here, represents a hypothesis, but not scientific consensus.
 * Sahelanthropus (?)†
 * Sahelanthropus tchadensis
 * Orrorin†
 * Orrorin tugenensis
 * Orrorin praegens
 * Ardipithecus†
 * Ardipithecus ramidus
 * Ardipithecus kadabba
 * Kenyanthropus†
 * Kenyanthropus platyops
 * Australopithecus†
 * Australopithecus bahrelghazali
 * Australopithecus anamensis
 * Australopithecus afarensis
 * Australopithecus africanus
 * Australopithecus garhi
 * Australopithecus sediba
 * Paranthropus†
 * Paranthropus aethiopicus
 * Paranthropus robustus
 * Paranthropus boisei
 * Homo – immediate ancestors of modern humans
 * Homo gautengensis† (probable H. habilis specimens)
 * Homo rudolfensis†
 * Homo habilis†
 * Homo floresiensis†
 * Homo erectus†
 * Homo ergaster†
 * Homo antecessor†
 * Homo heidelbergensis†
 * Homo cepranensis† (probable early H. sapiens specimens)
 * Denisovans (scientific name has not yet been assigned)†
 * Homo neanderthalensis†
 * Homo rhodesiensis† (probable late H. heidelbergensis specimens)
 * Homo sapiens
 * Anatomically modern human, Homo sapiens sapiens
 * Archaic Homo sapiens (Cro-magnon)†
 * Red Deer Cave people† (scientific name has not yet been assigned)
 * Homo sapiens idaltu† (classification not widely accepted)

Evolution
The age of the subfamily Homininae (of the Homininae–Ponginae last common ancestor) is estimated at some 14 to 12.5 million years (Sivapithecus). Its separation into Gorillini and Hominini (the "gorilla–human last common ancestor", GHLCA) is estimated to have occurred at about (TGHLCA) during the late Miocene, close to the age of Nakalipithecus nakayamai.

There is evidence there was interbreeding of Gorillas and the Pan–Homo ancestors until right up to the Pan–Homo split.

Evolution of bipedalism
Recent studies of Ardipithecus ramidus (4.4 million years old) and Orrorin tugenensis (6 million years old) suggest some degree of bipedalism. Australopithecus and early Paranthropus may have been bipedal. Very early hominins such as Ardipithecus ramidus may have possessed an arboreal type of bipedalism.

The evolution of bipedalism encouraged multiple changes among hominins especially when it came to bipedalism in humans as they were now able to do many other things as they began to walk with their feet. These changes included the ability to now use their hands to create tools or carry things with their hands, the ability to travel longer distances at a faster speed, and the ability to hunt for food. According to researchers, humans were able to be bipedalists due to Darwin's Principle of natural selection. Darwin himself believed that larger brains in humans made an upright gait necessary, but had no hypothesis for how the mechanism evolved.

The first major theory attempting to directly explanation the origins of bipedalism was the Savannah hypothesis (Dart 1925.) This theory hypothesized that hominins became bipedalists due to the environment of the Savanna such as the tall grass and dry climate. This was later proven to be incorrect due to fossil records that showed that hominins were still climbing trees during this era.

Anthropologist Owen Lovejoy has suggested that bipedalism was a result of sexual dimorphism in efforts to help with the collecting of food. In his Male Provisioning Hypothesis introduced in 1981, lowered birth rates in early hominids increased pressure on males to provide for females and offspring. While females groomed and cared for their children with the family group, males ranged to seek food and returned bipadally with full arms. Males who could better provide for females in this model were more likely to mate and produce offspring.

Anthropologist Yohannes Haile-Selassie, an expert on Australopithecus anamensis, discusses the evidence that Australopithecus were one of the first hominins to evolve into obligate bipedalists. The remains of this subfamily are very important in the field of research as it presents possible information regarding how these primates adapted from tree life to terrestrial life. This was a huge adaptation as it encouraged many evolutionary changes within hominins including the ability to use their hand to make tools and gather food, as well as a larger brain development due to their change in diet.

Brain size evolution
There has been a gradual increase in brain volume (brain size) as the ancestors of modern humans progressed along the timeline of human evolution, starting from about 600 cm3 in Homo habilis up to 1500 cm3 in Homo neanderthalensis. However, modern Homo sapiens have a brain volume slightly smaller (1250 cm3) than Neanderthals, women have a brain slightly smaller than men and the Flores hominids (Homo floresiensis), nicknamed hobbits, had a cranial capacity of about 380 cm3 (considered small for a chimpanzee), about a third of the Homo erectus average. It is proposed that they evolved from H. erectus as a case of insular dwarfism. In spite of their smaller brain, there is evidence that H. floresiensis used fire and made stone tools at least as sophisticated as those of their proposed ancestors H. erectus. In this case, it seems that for intelligence, the structure of the brain is more important than its size.

The current size of the human brain is a big distinguishing factor that separates humans from other primates. Recent examination of the human brain shows that the brain of a human is about more than four times the size of great apes and 20 times larger than the brain size of old world monkeys. A study was conducted to help determine the evolution of the brain size within the sub family Homininae that tested the genes ASPM (abnormal spindle-like microcephaly associated) and MCHP1 (microcephalin-1) and their association with the human brain. In this study researchers discovered that the increase in brain size is correlated to the increase of both ASP and MCPH1. MCPH1 is very polymorphic in humans compared to gibbons, Old World monkeys. This gene helps encourage the growth of the brain. Further research indicated that the MCPH1 gene in humans could have also been an encouraging factor of population expansion. Other researchers have included that the diet was an encouraging factor to brain size as protein intake increased this helped brain development.

Evolution of family structure and sexuality
Sexuality is related to family structure and partly shapes it. The involvement of fathers in education is quite unique to humans, at least when compared to other Homininae. Concealed ovulation and menopause in women both also occur in a few other primates however, but are uncommon in other species. Testis and penis size seems to be related to family structure: monogamy or promiscuity, or harem, in humans, chimpanzees or gorillas, respectively. The levels of sexual dimorphism are generally seen as a marker of sexual selection. Studies have suggested that the earliest hominins were dimorphic and that this lessened over the course of the evolution of the genus Homo, correlating with humans becoming more monogamous, whereas gorillas, who live in harems, show a large degree of sexual dimorphism. Concealed (or "hidden") ovulation means that the phase of fertility is not detectable in women, whereas chimpanzees advertise ovulation via an obvious swelling of the genitals. Women can be partly aware of their ovulation along the menstrual phases, but men are essentially unable to detect ovulation in women. Most primates have semi-concealed ovulation, thus one can think that the common ancestor had semi-concealed ovulation, that was inherited by gorillas, and that later evolved in concealed ovulation in humans and advertised ovulation in chimpanzees. Menopause also occurs in rhesus monkeys, and possibly in chimpanzees, but does not in gorillas and is quite uncommon in other primates (and other mammal groups).

title: Homininae


text:
Homo habilis

Homo habilis (lit. 'handy man') is an extinct species of archaic human from the Early Pleistocene of East and South Africa about 2.3 million years ago to 1.65 million years ago (mya). Upon species description in 1964, H. habilis was highly contested, with many researchers recommending it be synonymised with Australopithecus africanus, the only other early hominin known at the time, but H. habilis received more recognition as time went on and more relevant discoveries were made. By the 1980s, H. habilis was proposed to have been a human ancestor, directly evolving into Homo erectus, which directly led to modern humans. This viewpoint is now debated. Several specimens with insecure species identification were assigned to H. habilis, leading to arguments for splitting, namely into "H. rudolfensis" and "H. gautengensis" of which only the former has received wide support.

Like contemporary Homo, H. habilis brain size generally varied from 500–900 cm3. The body proportions of H. habilis are only known from two highly fragmentary skeletons, and is based largely on assuming a similar anatomy to the earlier australopithecines. Because of this, it has also been proposed H. habilis be moved to the genus Australopithecus as Australopithecus habilis. However, the interpretation of H. habilis as a small-statured human with inefficient long-distance travel capabilities has been challenged. The presumed female specimen OH 62 is traditionally interpreted as having been 100–120 cm in height and 20–37 kg in weight assuming australopithecine-like proportions, but assuming humanlike proportions she would have been about 148 cm and 35 kg. Nonetheless, H. habilis may have been at least partially arboreal like what is postulated for australopithecines. Early hominins are typically reconstructed as having thick hair and marked sexual dimorphism with males much larger than females, though relative male and female size is not definitively known.

H. habilis manufactured the Oldowan stone-tool industry and mainly used tools in butchering. Early Homo, compared to australopithecines, are generally thought to have consumed high quantities of meat and, in the case of H. habilis, scavenged meat. Typically, early hominins are interpreted as having lived in polygynous societies, though this is highly speculative. Assuming H. habilis society was similar to that of modern savanna chimpanzees and baboons, groups may have numbered 70–85 members. This configuration would be advantageous with multiple males to defend against open savanna predators, such as big cats, hyenas and crocodiles. H. habilis coexisted with H. rudolfensis, H. ergaster / H. erectus and Paranthropus boisei.

Research history
The first recognised remains—OH 7, partial juvenile skull, hand, and foot bones dating to 1.75 million years ago (mya)—were discovered in Olduvai Gorge, Tanzania, in 1960 by Jonathan Leakey. However, the actual first remains—OH 4, a molar—were discovered by the senior assistant of Louis and Mary Leakey (Jonathan's parents), Heselon Mukiri, in 1959, but this was not realised at the time. By this time, the Leakeys had spent 29 years excavating in Olduvai Gorge for early hominin remains, but had instead recovered mainly other animal remains as well as the Oldowan stone-tool industry. The industry had been ascribed to Paranthropus boisei (at the time "Zinjanthropus") in 1959 as it was the first and only hominin recovered in the area, but this was revised upon OH 7's discovery. In 1964, Louis, South African palaeoanthropologist Phillip V. Tobias, and British primatologist John R. Napier officially assigned the remains into the genus Homo, and, on recommendation by Australian anthropologist Raymond Dart, the specific name H. habilis, meaning "able, handy, mentally skillful, vigorous" in Latin. The specimen's association with the Oldowan (then considered evidence of advanced cognitive ability) was also used as justification for classifying it into Homo. OH 7 was designated the holotype specimen.

After description, it was hotly debated if H. habilis should be reclassified into Australopithecus africanus (the only other early hominin known at the time), in part because the remains were so old and at the time Homo was presumed to have evolved in Asia (with the australopithecines having no living descendants). Also, the brain size was smaller than what Wilfrid Le Gros Clark proposed in 1955 when considering Homo. The classification H. habilis began to receive wider acceptance as more fossil elements and species were unearthed. In 1983, Tobias proposed that A. africanus was a direct ancestor of Paranthropus and Homo (the two were sister taxa), and that A. africanus evolved into H. habilis which evolved into H. erectus which evolved into modern humans (by a process of cladogenesis). He further said that there was a major evolutionary leap between A. africanus and H. habilis, and thereupon human evolution progressed gradually because H. habilis brain size had nearly doubled compared to australopithecine predecessors.

Many had accepted Tobias' model and assigned Late Pliocene to Early Pleistocene hominin remains outside the range of Paranthropus and H. erectus into H. habilis. For non-skull elements, this was done on the basis of size as there was a lack of clear diagnostic characteristics. Because of these practices, the range of variation for the species became quite wide, and the terms H. habilis sensu stricto (i.e. strictly) and H. habilis sensu lato (i.e. broadly) were in use to include and exclude, respectively, more discrepant morphs. To address this, in 1985, English palaeoanthropologist Bernard Wood proposed that the comparatively massive skull KNM-ER 1470 from Lake Turkana, Kenya, discovered in 1972 and assigned to H. habilis, actually represented a different species, now referred to as Homo rudolfensis. It is also argued that instead it represents a male specimen whereas other H. habilis specimens are female. Early Homo from South Africa have variously been assigned to H. habilis or H. ergaster / H. erectus, but species designation has largely been unclear. In 2010, Australian archaeologist Darren Curoe proposed splitting off South African early Homo into a new species, "Homo gautengensis".

In 1986, OH 62, a fragmentary skeleton was discovered by American anthropologist Tim D. White in association with H. habilis skull fragments, definitively establishing aspects of H. habilis skeletal anatomy for the first time, and revealing more Australopithecus-like than Homo-like features. Because of this, as well as similarities in dental adaptations, Wood and biological anthropologist Mark Collard suggested moving the species to Australopithecus in 1999. However, reevaluation of OH 62 to a more humanlike physiology, if correct, would cast doubt on this. The discovery of the 1.8 Ma Georgian Dmanisi skulls in the early 2000s, which exhibit several similarities with early Homo, has led to suggestions that all contemporary groups of early Homo in Africa, including H. habilis and H. rudolfensis, are the same species and should be assigned to H. erectus.

Classification
There is still no wide consensus as to whether or not H. habilis is ancestral to H. ergaster / H. erectus or is an offshoot of the human line, and whether or not all specimens assigned to H. habilis are correctly assigned or the species is an assemblage of different Australopithecus and Homo species. Nonetheless, H. habilis and H. rudolfensis generally are recognised members of the genus at the base of the family tree, with arguments for synonymisation or removal from the genus not widely adopted.

Though it is now largely agreed upon that Homo evolved from Australopithecus, the timing and placement of this split has been much debated, with many Australopithecus species having been proposed as the ancestor. The discovery of LD 350-1, the oldest Homo specimen, dating to 2.8 mya, in the Afar Region of Ethiopia may indicate that the genus evolved from A. afarensis around this time. This specimen was initially classified as Homo sp., though subsequent studies have suggested that it also shares characteristics with Australopithecus and that it is clearly distinct from H. habilis. The oldest H. habilis specimen, A.L. 666-1, dates to 2.3 mya, but is anatomically more derived (has less ancestral, or basal, traits) than the younger OH 7, suggesting derived and basal morphs lived concurrently, and that the H. habilis lineage began before 2.3 mya. Based on 2.1-million-year-old stone tools from Shangchen, China, H. habilis or an ancestral species may have dispersed across Asia. The youngest H. habilis specimen, OH 13, dates to about 1.65 mya.

Skull
It has generally been thought that brain size increased along the human line especially rapidly at the transition between species, with H. habilis brain size smaller than that of H. ergaster / H. erectus, jumping from about 600–650 cc in H. habilis to about 900–1000 cc in H. ergaster and H. erectus. However, a 2015 study showed that the brain sizes of H. habilis, H. rudolfensis, and H. ergaster generally ranged between 500–900 cc after reappraising the brain volume of OH 7 from 647–687 cc to 729–824 cc. This does, nonetheless, indicate a jump from australopithecine brain size which generally ranged from 400–500 cc.

The brain anatomy of all Homo features an expanded cerebrum in comparison to australopithecines. The pattern of striations on the teeth of OH 65 slant right, which may have been accidentally self-inflicted when the individual was pulling a piece of meat with its teeth and the left hand while trying to cut it with a stone tool using the right hand. If correct, this could indicate right handedness, and handedness is associated with major reorganisation of the brain and the lateralisation of brain function between the left and right hemispheres. This scenario has also been hypothesised for some Neanderthal specimens. Lateralisation could be implicated in tool use. In modern humans, lateralisation is weakly associated with language.

The tooth rows of H. habilis were V-shaped as opposed to U-shaped in later Homo, and the mouth jutted outwards (was prognathic), though the face was flat from the nose up.

Build
Based on the fragmentary skeletons OH 62 (presumed female) and KNM-ER 3735 (presumed male), H. habilis body anatomy has generally been considered to have been more apelike than even that of the earlier A. afarensis and consistent with an at least partially arboreal lifestyle in the trees as is assumed in australopithecines. Based on OH 62 and assuming comparable body dimensions to australopithecines, H. habilis has generally been interpreted as having been small-bodied like australopithecines, with OH 62 generally estimated at 100–120 cm in height and 20–37 kg in weight. However, assuming longer, modern humanlike legs, OH 62 would have been about 148 cm and 35 kg, and KNM-ER 3735 about the same size. For comparison, modern human men and women in the year 1900 averaged 163 cm and 152.7 cm, respectively. It is generally assumed that pre-H. ergaster hominins, including H. habilis, exhibited notable sexual dimorphism with males markedly bigger than females. However, relative female body mass is unknown in this species.

Early hominins, including H. habilis, are thought to have had thick body hair coverage like modern non-human apes because they appear to have inhabited colder regions and are thought to have had a less active lifestyle than (presumed hairless) post-ergaster species. Consequently, they probably required thick body hair to stay warm. Based on dental development rates, H. habilis is assumed to have had an accelerated growth rate compared to modern humans, more like that of modern non-human apes.

Limbs
The arms of H. habilis and australopithecines have generally been considered to have been proportionally long and so adapted for climbing and swinging. In 2004, anthropologists Martin Haeusler and Henry McHenry argued that, because the humerus to femur ratio of OH 62 is within the range of variation for modern humans, and KNM-ER 3735 is close to the modern human average, it is unsafe to assume apelike proportions. Nonetheless, the humerus of OH 62 measured 258–270 mm long and the ulna (forearm) 245–255 mm, which is closer to the proportion seen in chimpanzees. The hand bones of OH 7 suggest precision gripping, important in dexterity, as well as adaptations for climbing. In regard to the femur, traditionally comparisons with the A. afarensis specimen AL 288-1 have been used to reconstruct stout legs for H. habilis, but Haeusler and McHenry suggested the more gracile OH 24 femur (either belonging to H. ergaster / H. erectus or P. boisei) may be a more apt comparison. In this instance, H. habilis would have had longer, humanlike legs and have been effective long-distance travellers as is assumed to have been the case in H. ergaster. However, estimating the unpreserved length of a fossil is highly problematic. The thickness of the limb bones in OH 62 is more similar to chimpanzees than H. ergaster / H. erectus and modern humans, which may indicate different load bearing capabilities more suitable for arboreality in H. habilis. The strong fibula of OH 35 (though this may belong to P. boisei) is more like that of non-human apes, and consistent with arboreality and vertical climbing.

OH 8, a foot, is better suited for terrestrial movement than the foot of A. afarensis, though it still retains many apelike features consistent with climbing. However, the foot has projected toe bone and compacted mid-foot joint structures, which restrict rotation between the foot and ankle as well as at the front foot. Foot stability enhances the efficiency of force transfer between the leg and the foot and vice versa, and is implicated in the plantar arch elastic spring mechanism which generates energy while running (but not walking). This could possibly indicate H. habilis was capable of some degree of endurance running, which is typically thought to have evolved later in H. ergaster / H. erectus.

Society
Typically, H. ergaster / H. erectus is considered to have been the first human to have lived in a monogamous society, and all preceding hominins were polygynous. However, it is highly difficult to speculate with any confidence the group dynamics of early hominins. The degree of sexual dimorphism and the size disparity between males and females is often used to correlate between polygyny with high disparity and monogamy with low disparity based on general trends (though not without exceptions) seen in modern primates. Rates of sexual dimorphism are difficult to determine as early hominin anatomy is poorly represented in the fossil record. In some cases, sex is arbitrarily determined in large part based on perceived size and apparent robustness in the absence of more reliable elements in sex identification (namely the pelvis). Mating systems are also based on dental anatomy, but early hominins possess a mosaic anatomy of different traits not seen together in modern primates; the enlarged cheek teeth would suggest marked size-related dimorphism and thus intense male–male conflict over mates and a polygynous society, but the small canines should indicate the opposite. Other selective pressures, including diet, can also dramatically impact dental anatomy. The spatial distribution of tools and processed animal bones at the FLK Zinj and PTK sites in Olduvai Gorge indicate the inhabitants used this area as a communal butchering and eating grounds, as opposed to the nuclear family system of modern hunter gatherers where the group is subdivided into smaller units each with their own butchering and eating grounds.

The behaviour of early Homo, including H. habilis, is sometimes modelled on that of savanna chimps and baboons. These communities consist of several males (as opposed to a harem society) in order to defend the group on the dangerous and exposed habitat, sometimes engaging in a group display of throwing sticks and stones against enemies and predators. The left foot OH 8 seems to have been bitten off by a crocodile, possibly Crocodylus anthropophagus, and the leg OH 35, which either belongs to P. boisei or H. habilis, shows evidence of leopard predation. H. habilis and contemporary hominins were likely predated upon by other large carnivores of the time, such as (in Olduvai Gorge) the hunting hyena Chasmaporthetes nitidula, and the saber-toothed cats Dinofelis and Megantereon. In 1993, American palaeoanthropologist Leslie C. Aiello and British evolutionary psychologist Robin Dunbar estimated that H. habilis group size ranged from 70–85 members—on the upper end of chimp and baboon group size—based on trends seen in neocortex size and group size in modern non-human primates.

H. habilis coexisted with H. rudolfensis, H. ergaster / H. erectus, and P. boisei. It is unclear how all of these species interacted. To explain why P. boisei was associated with Olduwan tools despite not being the knapper (the one who made the tools), Leakey and colleagues, when describing H. habilis, suggested that one possibility was P. boisei was killed by H. habilis, perhaps as food. However, when describing P. boisei five years earlier, Louis Leakey said, "There is no reason whatever, in this case, to believe that the skull represents the victim of a cannibalistic feast by some hypothetical more advanced type of man."

Diet
It is thought H. habilis derived meat from scavenging rather than hunting (scavenger hypothesis), acting as a confrontational scavenger and stealing kills from smaller predators such as jackals or cheetahs. Fruit was likely also an important dietary component, indicated by dental erosion consistent with repetitive exposure to acidity. Based on dental microwear-texture analysis, H. habilis (like other early Homo) likely did not regularly consume tough foods. Microwear-texture complexity is, on average, somewhere between that of tough-food eaters and leaf eaters (folivores), and points to an increasingly generalised and omnivorous diet. Freshwater fish likely were also consumed, evidenced by the findings of fish remains at archaeological sites most likely associated with H. habilis.

It is typically thought that the diets of H. habilis and other early Homo had a greater proportion of meat than Australopithecus, and that this led to brain growth. The main hypotheses regarding this are: meat is energy- and nutrient-rich and put evolutionary pressure on developing enhanced cognitive skills to facilitate strategic scavenging and monopolise fresh carcasses, or meat allowed the large and calorie-expensive ape gut to decrease in size allowing this energy to be diverted to brain growth. Alternatively, it is also suggested that early Homo, in a drying climate with scarcer food options, relied primarily on underground storage organs (such as tubers) and food sharing, which facilitated social bonding among both male and female group members. However, unlike what is presumed for H. ergaster and later Homo, short-statured early Homo are generally considered to have been incapable of endurance running and hunting, and the long and Australopithecus-like forearm of H. habilis could indicate early Homo were still arboreal to a degree. Also, organised hunting and gathering is thought to have emerged in H. ergaster. Nonetheless, the proposed food-gathering models to explain large brain growth necessitate increased daily travel distance. It has also been argued that H. habilis instead had long, modern humanlike legs and was fully capable of effective long distance travel, while still remaining at least partially arboreal.

Large incisor size in H. habilis compared to Australopithecus predecessors implies this species relied on incisors more. The bodies of the mandibles of H. habilis and other early Homo are thicker than those of modern humans and all living apes, more comparable to Australopithecus. The mandibular body resists torsion from the bite force or chewing, meaning their jaws could produce unusually powerful stresses while eating. The greater molar cusp relief in H. habilis compared to Australopithecus suggests the former used tools to fracture tough foods (such as pliable plant parts or meat), otherwise the cusps would have been more worn down. Nonetheless, the jaw adaptations for processing mechanically challenging food indicates technological advancement did not greatly affect diet.

Technology
H. habilis is associated with the Early Stone Age Oldowan stone tool industry. Individuals likely used these tools primarily to butcher and skin animals and crush bones, but also sometimes to saw and scrape wood and cut soft plants. Knappers - individuals shaping stones - appear to have carefully selected lithic cores and knew that certain rocks would break in a specific way when struck hard enough and on the right spot, and they produced several different types, including choppers, polyhedrons, and discoids. Nonetheless, specific shapes were likely not thought of in advance, and probably stem from a lack of standardisation in producing such tools as well as the types of raw materials at the knappers' disposal. For example, spheroids are common at Olduvai, which features an abundance of large and soft quartz and quartzite pieces, whereas Koobi Fora lacks spheroids and provides predominantly hard basalt lava rocks. Unlike the later Acheulean culture invented by H. ergaster / H. erectus, Oldowan technology does not require planning and foresight to manufacture, and thus does not indicate high cognition in Oldowan knappers, though it does require a degree of coordination and some knowledge of mechanics. Oldowan tools infrequently exhibit retouching and were probably discarded immediately after use most of the time.

The Oldowan was first reported in 1934, but it was not until the 1960s that it become widely accepted as the earliest culture, dating to 1.8 mya, and as having been manufactured by H. habilis. Since then, more discoveries have placed the origins of material culture substantially backwards in time, with the Oldowan being discovered in Ledi-Geraru and Gona in Ethiopia dating to 2.6 mya, perhaps associated with the evolution of the genus. Australopithecines are also known to have manufactured tools, such as the 3.3 Ma Lomekwi stone tool industry, and some evidence of butchering from about 3.4 mya. Nonetheless, the comparatively sharp-edged Oldowan culture was a major innovation from australopithecine technology, and it would have allowed different feeding strategies and the ability to process a wider range of foods, which would have been advantageous in the changing climate of the time. It is unclear if the Oldowan was independently invented or if it was the result of hominin experimentation with rocks over hundreds of thousands of years across multiple species.

In 1962, a 12x14x1 ft circle made with volcanic rocks was discovered in Olduvai Gorge. At 2–2.5 ft intervals, rocks were piled up to 6–9 in high. Mary Leakey suggested the rock piles were used to support poles stuck into the ground, possibly to support a windbreak or a rough hut. Some modern-day nomadic tribes build similar low-lying rock walls to build temporary shelters upon, bending upright branches as poles and using grasses or animal hide as a screen. Dating to 1.75 mya, it is attributed to some early Homo, and is the oldest-claimed evidence of architecture.

title: Homo habilis