This is equivalent to the condition that every element has at least one inverse and that the idempotents of commute Lawson Note that if is an inverse of , then is an idempotent. Clifford, A. The Algebraic Theory of Semigroups, Vol. Providence, RI: Amer.

Lawson, M. Singapore: World Scientific, In , Sushkevich published a monograph on his work up to that time: Theory of Generalized Groups in Russian. Except for in a few pages of his monograph, all his generalized groups are associative. Sushkevich continued publishing work on semigroup theory through , but his work failed to have a major impact on the theory in the U.

In part this may have been because much of his work was published in journals based in Kharkov and the difficulty in getting access to many regional journals in the U.

The development of algebraic semigroup theory in the U. Even as he was finishing his thesis Clifford was working on another semigroup problem, one on generalization of group axiomatics. In , G. Hollings discusses the Rees Theorem in detail in Chapter 6. Rees subsequently published four more papers on semigroups, but he abandoned the study of semigroups in the late s and moved to commutative ring theory.

## ALGEBRAIC COMBINATORICS

Several others who were or became semigroup researchers of note were at Bletchley Park at various times during the war. This included Preston, Green, and a U. Navy officer named Alfred H. Clifford, who was at Bletchley Park from May to June This paper was very influential in France and world-wide. The French school of semigroup theory continued to thrive for decades.

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Hollings discusses these in some detail, as well as the interactions between schools. The most influential mathematician was G. Lyapin — He became interested in semigroups about and completed a doctoral dissertation on the subject at Leningrad State University in Elements of an abstract theory of systems with one operation.

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Lyapin had been teaching at Leningrad State University since , but in the Marxist-Leninist ideologues at the University declared semigroup theory to be one of three mathematical topics that were ideologically objectionable. Lyapin continued his research, however, taking a teaching job at Leningrad State Pedagogical Institute.

He became the most influential semigroup theorist in the U. Green, Gordon Preston, and Douglas Munn. Green obtained the Ph. Preston burst on the semigroup scene in with the publication of a series of three papers on inverse semigroups. Wagner Saratov, U. R had published some similar work in Inverse semigroups rapidly became one of the most important topics in semigroup theory, both in the East and the West. Two monographs on the subject [9] and [11] have appeared.

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Walter Douglas Munn received the Ph. He found out about inverse semigroups from conversations with Preston. His first publication was on semigroup algebras, in His next was on matrix representations of semigroups, in Many other semigroup papers by Munn followed, not only on these two topics, but on inverse semigroups and allied topics. Schwarz originally worked in other areas of algebra and his first semigroup paper appeared in In the s he published seminal and deep work on ideals in semigroups and on periodic semigroups. These ideas were picked up by others of the Slovak school and carried further, e.

Takayuki Tamura — stands out as the founder and leader of the Japanese school of semigroup research.

## The Algebraic Theory of Semigroups - Arthur Hoblitzelle Clifford, G. B. Preston - Google Books

His first paper on semigroup theory appeared in These topics were picked up by others in the Japanese school, some of them working jointly with Tamura. The notion of semilattice decompositions first appeared in work by Tamura and Naoki Kimura in the mids. Tamura and his colleagues did immense computational work on enumerating the number of non-isomorphic, non-anti-isomorphic semigroups of a given order.

This problem was also being attacked actively in the U. For details on this, see pp. Margolis and J.

### SEMIGROUPS GENERATED BY PARTITIONS

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Pin and C.

Reiterman , The Birkhoff theorem for finite algebras , Algebra Universalis , vol. Ch and.