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?