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A Horn clause in the language of a universal algebra is a disjunction of equations and of at most one inequality ("equation" and "inequality" are the terms used by A.Horn in his paper "On sencences which are true of direct unions of algebras", Journal of Symbolic Logic, 1951, 16(1), 14-21.). Quasi-identities and, in particular, identities are (equivalent to) Horn clauses, and the classes of algebras I know are axiomatized only by these.

Is there any research of interesting classes of algebras axiomatized by non-Horn clauses? I believe, the algebraic methods and structures used in such research, would be useful in algebraization of set theory. Currently, there is an "Algebraic set theory" (AST) in category theory presentation, according to which two operationsare foundational for set theory. I am working on a presentation of set theory with one binaryfoundational operation playing the role of membership relationship as discussed in this question. I doubt, that AST can be presented with axioms only in Horn clause form, and I am quite sure some the axioms in my approach cannot be so presented.

For example, I seriously doubt that the axiom of "atomicity" in this question of mine, can be presented as a Horn clause.

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You can use the term "algebra" however you want. There are various contexts where it may have more specific meanings, but it is also a widely used generic term for "some kind of algebraic structure". –  Eric Wofsey Aug 2 at 23:19
    
Are there also other kinds of formulas except identities and quasi-identities specific to algebraic research? –  Ioachim Drugus Aug 3 at 12:02
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There are also pseudoidentities ( for finite structures) and algebras with a discriminator term (which sometimes form varieties but start with a nonequational definition), but there are also studies involving other kinds of constructions which don't have associated preservation theorems. Without more motivation or specification, the question becomes too open ended for this forum. Also, I suspect the relevant area may be algebraic logic more than general algebra. –  The Masked Avenger Aug 4 at 11:12
    
I am now looking into pseudoidentities and algebras with a discriminator term and they really sound on topic and useful - thank you. Probably, algebraic logic is more relevant than general algebra, but both I did not find a tag like "algebraic logic", and I believe, that in abstract algebra it might be easier to find concrete results which would help figure out an approach to my "adjunction algebras". –  Ioachim Drugus Aug 4 at 23:36

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