Inverse Monoids with a Natural Semilattice Ordering

An inverse semigroup is a semigroup S such that for each x e S there exists a unique inverse x~ & S such that both xx~x=x and x~xx~=x~\ This condition is equivalent to S both being (von Neumann) regular and having all idempotents commute. The classic example of such a semigroup is the symmetric inverse semigroup Ix of all partial bijections on a set X under the standard composition of partial functions on X. More generally, for any reasonably endowed mathematical object Q, the set /n of all partial symmetries of Q (bijections between subobjects of Q respecting all relevant structure) forms an inverse semigroup, the symmetric inverse semigroup of Q. Nearly all such semigroups, however, are more than mere inverse semigroups. Clearly such semigroups possess both an identity 1 and a zero 0, so that one has at least inverse monoids with zero; but of greater consequence is the fact that the natural partial order on the semigroup (given by x ^y if and only if y = yy~x =xy~y) usually possesses infima of arbitrary non-empty subsets. When this occurs, every element x has a fixed point idempotent f[x] = 1 AX which is maximal among those idempotents lying beneath x in the natural partial order. (For example, in the symmetric inverse semigroup on a set X, infima are given by intersections, inf,e/ OLJ = H/e/ h while the fixed point idempotent f[a] of a partial bijection a is the identity map of its set of fixed points.) What is more, multiplication always distributes over the induced meet operation so that one obtains a type of semiring structure. These observations are an indication that the class of inverse monoids which are also meet semilattices with respect to their natural partial order are worthy of further investigation. It is the purpose of this paper to initiate just such a study. To do so, it will be convenient to have a single term to denote this class of algebraic structures. The term adopted in this paper is inverse algebra with 'algebra' being understood in the sense of 'universal algebra'. The term should not be confused with the semigroup algebra k[S] of an inverse semigroup S over a field. The paper is divided into six sections, with the first presenting examples and results which are foundational to the rest of the paper. Congruences are considered in the next section, where congruences of inverse algebras are characterized as those monoid congruences that are generated from their restriction to the subalgebra of idempotents. Section 3 studies a number of universal constructions including the (relatively) free generation of inverse algebras from inverse monoids and completions of inverse algebras. The structure and consequent classification of monogenic inverse algebras are given