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When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in many proofs (like behaving well with respect to morphisms and quotients)?

For instance, is the following axiom acceptable, in a structure equipped with a unary function $f$ and a binary operator $\circ$: $$\forall x, \text{ if }x\circ x=x\text{ then } f(x)=x.$$

More specifically, does it make sense to study varieties generated by (subclasses of) such classes of objects, even if the class of structures so defined is not a variety, since this axiom is not an equation?

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*** Be my guest. *** –  Wlodzimierz Holsztynski Jul 24 '13 at 16:33
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In the model theory of first-order logic, of course any kind of axioms are allowed and studied. Yet, often one can say much more when the axiomatizations have a special form. In universal algebra, for example, one often fruitfully restricts to equational axiomatizations, leading to the concept of a variety: see en.wikipedia.org/wiki/Universal_algebra#Varieties –  Joel David Hamkins Jul 24 '13 at 16:47
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There is a descriptive term which I have forgotten which I will call x. It might be "representation" but I don't think so. Birkhoff's HSP theorem is an x theorem, as is the one which says the class of models axiomatized by Horn sentences (eq. 1 implies eq. 2) is a quasivariety, a class closed under cetain algebraic constructions. There are many x theorems in Model Theory. Solve for x and I think you will have a general answer to your question. –  The Masked Avenger Jul 24 '13 at 17:02
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The specific example you gave is an example of what category theorists call an essentially algebraic theory, where partially defined operations (with domains specified as loci of equations of previously given operations) may be admitted. The theory of categories is an example of an essentially algebraic theory. Categories of models of (finitary) essentially algebraic theories are characterized as locally finitely presentable categories. See the book on Locally Presentable and Accessible Categories by Adamek and Rosicky. –  Todd Trimble Jul 24 '13 at 20:13
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I am a little surprised this was shut down so quickly. Meta: meta.mathoverflow.net/a/538/2926 –  Todd Trimble Jul 24 '13 at 20:20
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2 Answers

This is what is known as an equational Horn clause. It is an implication between two equations. It is a special case of what is called a near equational theory, which is one with operations and partial operations, the latter having domains given by equations using the ordinary (total) operations, plus equations involving the partial and total operations. There is an apparent generalization in which you allow partial operations whose domains are given by equations involving partial operations, but it turns out not to be more general.

A good example is the category of (small) categories in which the domain of the composition operation is given equationally in terms of the total operations of domain and codomain. This is discussed in detail in ``Category Theory for Computing Science'', available free at http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf.

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x = "preservation".

Usually one is motivated to study a structure because it serves as a model for something of interest. Likewise a certain statement or class of statements may have nice consequences for its class of models. It is uncommon to say 'I want to build this structure, but only if I can characterize it with regular identities or in iambic pentameter." Preservation theorems are the family of theorems that relate the shape of a characterizing language/theory to the shape of the constructions preserved.

If you need a special kind of suit to dress up a class of models of which your structure is a member, you may not get a precise fit: some members of the class are excluded, or perhaps some additional members are included. Whether this is desirable depends on the party for which one is dressing up.

The short answer is : it depends. If you are looking at alternative axiomatizations for your class, then why? Do you need a term rewriting system to work on it? Is recursive axiomatizability sufficient? Do you need a mixed presentation of a second order axiom and one or two modifying first order axioms? Are you trying to fit your class in some poset of defined classes?

Shape does matter. I can't tell you why before you tell me enough motivation.

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That said, we could use more descriptions in iambic pentameters... –  Mariano Suárez-Alvarez Sep 29 '13 at 0:40
    
the motivation is to generalize theorems of varieties of monoid of Reiterman and Eilenberg to more general structures. I don't know whether structures with this kind of axioms are valid, or if I have to restrict to structures defined only via equations without implication. Even more precisely, I want monoids to be quipped with an explicit operator omega, which is required to correspond to idempotent power in finite structures. –  Denis Jan 25 at 19:40
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