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

Inverse monoids with a natural semilattice ordering

We show that a topological semigroup of finite partial bijections $\mathscr{I}_\lambda^n$ of an infinite set with a compact subsemigroup of idempotents is absolutely $H$-closed and any countably compact topological semigroup does not contain $\mathscr{I}_\lambda^n$ as a subsemigroup. We give sufficient conditions onto a topological semigroup $\mathscr{I}_\lambda^1$ to be non-$H$-closed. Also we describe the structure of countably compact Brandt $\lambda^0$-extensions of topological monoids and study the category of countably compact Brandt $\lambda^0$-extensions of topological monoids with zero.

Topological semigroups of matrix units and countably compact Brandt $\lambda^0$-extensions of topological semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations $\mathscr{I}_\lambda^n$ of the rank $\leqslant n$ is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.

Semigroup Closures of Finite Rank Symmetric Inverse Semigroups

Rings and sheaves

We establish topological properties of the symmetric inverse topological semigroup of finite transformations of the rank ≤ n. We show that the topological inverse semigroup is algebraically h -closed in the class of topological inverse semigroups. Also we prove that a topological semigroup S with countably compact square S×S does not contain the semigroup for infinite cardinal λ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations of the rank ≤ n is the trivial semigroup.

Symmetric inverse topological semigroups of finite rank ≤ n

Epimorphisms of inverse semigroups of homeomorphisms between closed subsets

The symbol I G ( X ) denotes the inverse semigroup, under composition of functions, of all homeomorphisms between open subsets of a T 1 topological space X . The first result is that two such semigroups I G ( X ) and I G ( Y ) are isomorphic if and only if the spaces X and Y are homeomorphic. Ideals of I G ( X ) are next examined and it is shown that for many spaces X the semigroup I G ( X ) is 0-simple. We also look at congruences on I G ( X ); one result is that we determine a congruence which in many instances is the largest proper congruence on I G ( X ).

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Inverse semigroups of homeomorphisms between open subsets

Embedding inverse semigroups of homeomorphisms on closed subsets

If X is a topological space then S(X) will denote the semigroup, under composition, of all continuous functions from X into X  An element f in a semigroup is regular if there is an element g such that fgf = f  The regular elements of S(X) will be denoted by R(X)  Elements f and g are inverses of each other if fgf = f and gfg = g  Every regular element has an inverse [1] If every element in a semigroup has a unique inverse then the semigroup is an inverse semigroup In this paper we examine maximal inverse subsemigroups of S ( X )

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Bridget B. Baird

Papers

Epimorphisms of inverse semigroups of homeomorphisms between closed subsets

Inverse semigroups of homeomorphisms between open subsets

Embedding inverse semigroups of homeomorphisms on closed subsets

Maximal inverse subsemigroups of S ( X )

Green'sR-relation and climbing mountains