Birkhoff's variety theorem for relative algebraic theories

An algebraic theory, sometimes called an equational theory, is a theory defined by finitary operations and equations, such as the theories of groups and of rings. It is well known that algebraic theories are equivalent to finitary monads on $\mathbf{Set}$. In this paper, we generalize this phenomenon to locally finitely presentable categories using partial Horn logic. For each locally finitely presentable category $\mathscr{A}$, we define an "algebraic concept" relative to $\mathscr{A}$, which will be called an $\mathscr{A}$-relative algebraic theory, and show that $\mathscr{A}$-relative algebraic theories are equivalent to finitary monads on $\mathscr{A}$. Finally, we generalize Birkhoff's variety theorem for classical algebraic theories to our relative algebraic theories.Comment: 34 page

The number of clones determined by disjunctions of unary relations

We consider finitary relations (also known as crosses) that are definable via finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite parameter set $\Gamma$. We prove that whenever $\Gamma$ contains at least one non-empty relation distinct from the full carrier set, there is a countably infinite number of polymorphism clones determined by relations that are disjunctively definable from $\Gamma$. Finally, we extend our result to finitely related polymorphism clones and countably infinite sets $\Gamma$.Comment: manuscript to be published in Theory of Computing System