Abstract. An EQ-monoid $A$ is a monoid with distinguished subsemilattice $L$ with $1\in L$ and such that any $a,b\in A$ have a largest right equalizer in $L$. The class of all such monoids equipped with a binary operation that identifies this largest right equalizer is a variety. Examples include Heyting algebras, Cartesian products of monoids with zero, as well as monoids of relations and partial maps on sets. The variety is 0-regular (though not ideal determined and hence congruences do not permute), and we describe the normal subobjects in terms of a global semilattice structure. We give representation theorems for several natural subvarieties in terms of Boolean algebras, Cartesian products and partial maps. The case in which the EQ-monoid is assumed to be an inverse semigroup with zero is given particular attention. Finally, we define the derived category associated with a monoid having a distinguished subsemilattice containing the identity (a construction generalising the idea of a monoid category), and show that those monoids for which this derived category has equalizers in the semilattice constitute a variety of EQ-monoids.

AMS Subject Classification (1991): 06F05, 08A99, 08B05, 20M20

Received September 15, 2003, and in final form June 2, 2006. (Registered under 5932/2009.)