Carl Eberhart

One-parameter inverse semigroups

This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroup in a locally compact topological inverse semigroup. In it we characterize all one-parameter inverse semigroups. In order to accomplish this, we construct the free one-parameter inverse semigroups and then describe their congruences. 0. Let G be a subgroup of the multiplicative group of positive real numbers and let P denote the subsemigroup of G consisting of all x E G with x ? 1. Denote by Wp the class of all inverse semigroups H for which there isa homomorphism f: P -> H such that f(P) generates H (no proper inverse subsemigroup of H contains f(P)). We shall call such semigroups H one-parameter inverse semigroups and denote by W= UP Wp the class of all one-parameter inverse semigroups. The class W contains well-known semigroups. For example, each homomorphic image of a subgroup of R, the positive real numbers, is a member of W. Also the bicyclic semigroup B is a member of W, as is seen by noting that B is generated by a copy of the nonnegative integers. Indeed, if H is any elementary inverse semigroup, then H1 is generated by a homomorphic image of the nonnegative integers, and so is a one-parameter inverse semigroup. The main purpose of this paper is to describe all one-parameter inverse semigroups. In the process of doing this, we shall construct what we term the free oneparameter inverse semigroups Fp, one for each subgroup G of R and its associated semigroup P. The semigroup Fp is the only inverse semigroup (up to isomorphism) generated by a subsemigroup isomorphic with P which has the property that each homomorphism f: P -> S, an inverse semigroup, extends uniquely to a homomorphism f: Fp -> S. In particular, every H E Wp is a homomorphic image of Fp. We thus adopt the point of view that by describing Fp and the lattice of congruences of Fp for arbitrary P, we will have described all one-parameter inverse semigroups. We shall assume a certain familiarity with the algebraic theory of semigroups, particularly inverse semigroups. (See Clifford and Preston [1].) The existence and uniqueness of Fp is a consequence of a theorem due to McAlister [3, Theorem 33]. We were greatly aided in the actual description of Fp Presented to the Society, August 27, 1969; received by the editors June 10, 1969. AMS 1970 subject classifications. Primary 20M10; Secondary 20M05, 22A15.

Semimodularity in lattices of congruences

Abstract Necessary and sufficient conditions are given for the lattice of congruences on a band, normal band, band of groups, or normal band of groups to be semimodular. Also sufficient conditions are given for a general lattice to be semimodular and for a lattice of congruences on any semigroup to be semimodular.

CONGRUENCES ON AN ORTHODOX SEMIGROUP VIA THE MINIMUM INVERSE SEMIGROUP CONGRUENCE

It is well known that the lattice Λ( S ) of congruences on a regular semigroup S contains certain fundamental congruences. For example there is always a minimum band congruence β, which Spitznagel has used in his study of the lattice of congruences on a band of groups [16]. Of key importance to his investigation is the fact that β separates congruences on a band of groups in the sense that two congruences are the same if they have the same meet and join with β. This result enabled him to characterize θ-modular bands of groups as precisely those bands of groups for which ρ(ρ∨β, ρ∧β)is an embedding of Λ( S ) into a product of sublattices.

The shape classification of Torus-like and (n-sphere)-like continua

Abstract Let F be the collection of finite dimensional tori, F n the tori of dimension n or less, and S n the n -sphere. A complete shape classification for F -like Hausdorff continua, F n . like Hausdorff continua, and S n -like Hausdorff continua is obtained. If M is a F -like continuum, then there exists a unique compact connected abelian group having the same shape as M . If M is an S n -like Hausdorff continuum, then there exists a unique compact connected abelian group G M of dimension 1 or less such that the ( n −1)-fold suspension of G M has the same shape as M . As a consequence of these results, every F n -like Hausdorff continuum and every S n -like Hausdorff continuum has the shape of a metric continuum.

UNIVERSAL MAPS ON TREES

A map f : R → S of continua R and S is called a universal map from R to S if for any map g : R → S, f(x) = g(x) for some point x ∈ R. When R and S are trees, we characterize universal maps by reducing to the case of light minimal universal maps. The characterization uses the notions of combinatorial map and folded subedge of R.

The lattice of Knaster continua

Abstract It is shown that the set of Knaster continua possesses a natural lattice ordering. A description of this lattice is developed, and a series of questions is posed.