# Examples of algebraic structures that live at the intersection of varieties and antivarieties?

In Universal Algebra, by an equational variety of algebras we mean the class of all algebraic structures of a given signature satisfying certain (conjunctions of) identities. Dually, an antivariety satisfies certain disjunctions of negated equations (a.k.a. anti-identities). Moreover, by a disjunction of polynomial equations (DE) we mean a disjunction of identities. Obviously, an identity is a particular (degenerate) case of DE. Furthermore, one might note that anti-identities and DEs are, respectively, particular cases of Horn clauses and dual Horn clauses.

Some well-known classes of algebraic structures are very naturally characterized by a set of axioms containing both anti-identities and DEs. For instance, field axioms include an anti-identity (the 'zero-one law'), and a number of DEs, including a non-degenerate one that consists in a disjunction that deals with the multiplicative inverses of non-zero elements ('for any given field element, either it is zero or it has a multiplicative inverse'). If one generalizes the above definitions for signatures having relational symbols in the natural way (having positive literals replace identities, and negative literals replace anti-identities), other more or less natural examples of structures whose axiomatizations mix generalized anti-identities and generalized DEs may be found in Graph Theory (an example: connected oriented graphs).

Are there other well-known 'natural' and 'useful' examples of algebraic structures axiomatized by both DEs and anti-identities, in Mathematics or Computer Science?

• The first-order theory of graphs is not an algebraic structure, so it's not a good example. Also a disjunction of equations is not a particular case of a Horn clause. – Rob Arthan Apr 12 '17 at 20:58
• Thanks for the comment, @RobArthan. I will omit the example from Graph Theory, as it indeed requires the signature to be augmented with relation symbols. As for DEs not being a particular case of a Horn clause, I had meant to write dual Horn clauses (i.e., "disjunctions of literals with at most one negated literal"), exactly as in the link that I had added from the beginning. Shall fix that. Thanks again. – J Marcos Apr 12 '17 at 21:22
• Fixed, @RobArthan. Note that I ended up not omitting the example from Graph Theory, but rather modifying it minimally to have it making sense (I do hope it is clear now!). – J Marcos Apr 12 '17 at 21:29