Preface. Part A: Semigroups with Certain Types of Subsemigroup Lattices. I. Preliminaries. II. Semigroups with Modular or Semimodular Subsemigroup Lattices. III. Semigroups with Complementable Subsemigroups. IV. Finiteness Conditions. V. Inverse Semigroups with Certain Types of Lattices of Inverse Subsemigroups. VI. Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups. Part B: Properties of Subsemigroup Lattices. VII. Lattice Characteristics of Classes of Semigroups. VIII. Embedding Lattices in Subsemigroup Lattices. Part C: Lattice Isomorphisms. IX. Preliminaries on Lattice Isomorphisms. X. Cancellative Semigroups. XI. Commutative Semigroups. XII. Semigroups Decomposable into Rectangular Bands. XIII. Semigroups Defined by Certain Presentations. XIV. Inverse Semigroups. Bibliography. Index. List of Notations. List of Subsections Containing Unsolved Problems or Open Questions.

Semigroups and their subsemigroup lattices

Inverse semigroups determined by their bundles of correspondences

Let S and S' be two semigroups, each contained in a locally compact group. Under certain conditions on S and S' , we shall characterize those isometric additive surjections T: M(S) -* M(S') which preserve convolution squares. Our results generalize the classical results of Wendel and of Johnson and also Patterson's characterization of isometric involutions on measure algebras. Let G be a locally compact group, and let M(G) be the convolution measure algebra consisting of all regular complex Borel measures on G (J. L. Taylor [13]). For each subsemigroup S of G, let M(S) be the set of all ,u E M(G) that are concentrated on S. Then M(S) forms a norm-closed subalgebra of M(G). Under certain conditions on two semigroups S and S', each contained in a locally compact group, we shall characterize those isometric additive surjections T: M(S) -+ M(S') which preserve convolution squares: T(u *,u) = Tu * Tu . Such a T is either an isomorphism or an anti-isomorphism (as a mapping between two rings), and also either linear or conjugately linear (Theorem 7). Our characterization of such mappings generalizes not only the classical results of J. G. Wendel [14] and of B. E. Johnson [4] about isometric isomorphisms on measure algebras, but also P. L. Patterson's characterization of isometric involutions on measure algebras [7]. Our main result (Theorem 4) appears to be new even for finite groups. Recall that a semigroup S is said to satisfy the cancellation law if x $ y in S implies ax # ay and xa $4 ya for each a E S. Let f: S -S' be a mapping from a semigroup (or a ring) into another. We say that f is a semihomomorphism if it is either a homomorphism or an anti-homomorphism (f(xy) = f(y)f(x)). If, in addition, f is a bijection, we call f a semiisomorphism. We begin with a purely algebraic result, which will play central role in our work. Proposition 1. Let S be a semigroup, let S' be a semigroup satisfying the cancellation law, and let x x' : S -* S' be a mapping such that for each x and y e S, either (xy)' = x'y' or (xy)' = y'x'. Then x -* x' is a semihomomorphism. Received by the editors November 9, 1993 and, in revised form, May 30, 1994. 1991 Mathematics Subject Classification. Primary 43A10; Secondary 20M1 5.

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On square-preserving isometries of convolution algebras

Correspondences of completely regular semigroups and -isomorphisms of semigroups

Lev Naumovich Shevrin: fifty years in the service of mathematics

It is proved in this article that every half-isomorphism of a cancellative semigroup onto an arbitrary semigroup is either an isomorphism or an anti-isomorphism.

http://iopscience.iop.org/article/10.1070/IM1967v001n05ABEH000592/pdf

L N Ševrin

Papers

Half-isomorphisms of cancellative semigroups