The Algebraic Theory of Semigroups, Volume I by A. H. Clifford

By A. H. Clifford

The cloth during this quantity was once awarded in a second-year graduate path at Tulane college, in the course of the educational 12 months 1958-1959. The publication goals at being principally self-contained, however it is thought that the reader has a few familiarity with units, mappings, teams, and lattices. in simple terms in bankruptcy five will extra initial wisdom be required, or even there the classical definitions and theorems at the matrix representations of algebras and teams are summarized.

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The results just described were first found by Frobenius (Vber endliche Gruppen, Sitzungsber. Preuss. Akad. Wiss. Berlin, 1895, pp. 163-194), not for single elements of a semigroup, but for subsets (complexes) of a group (see Exercise 2 below). They were also found b y : Morgan Ward in 1933 (unpublished); Suschkewitsch [1937], Chapter 2, §19; Poole [1937]; Rees [1940]; and Climescu [1946]. We formulate them in the following theorem. 9. Let a be an element of a semigroup 8, and let be the cyclic subsemigroup of 8 generated by a.

If we define a product in CI by a]8 = y iiSaSp c Sy, then Q becomes thereby a band. We say that 8 is the union of the band Q of semigroups Sa (aGii). 5). Conversely, if is a homomorphism of a semigroup S upon a band Q, then the inverse image 8a = a-1 of each element a of £2 is a subsemigroup of 8, and S is the union of the band Q of semigroups Sa (aetl). If £L is commutative, we say that S is the union of the semilattice £1 of semigroups Sa (OCGQ). 6 asserts that S is the union of the semilattice E of semigroups Se (eeE).

I t will not be used in this book. If an element a of a semigroup S has an inverse in S, then a is evidently regular. 14) was noted by Thierrin [1952a]. Thus a regular semigroup is one in which every element has at least one inverse. 14. If a is a regular element of a semigroup S, say axa = a with x in S, then a has at least one inverse in S, in particular xax. PROOF. Let b = xax. Then aba — a(xax)a = ax(axa) = axa = a, bab = (xax)a(xax) = x(axa)(xax) = xa(xax) = x(axa)x = xax = 6. Hence b is an inverse of a.

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