Elsevier

Theoretical Computer Science

Volume 108, Issue 1, 1 February 1993, Pages 151-171
Theoretical Computer Science

Standard automata and semidirect products of transformation semigroups

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Abstract

We propose a decomposition of transformation semigroups (X, S) on a finite set X that provides

  • 1.

    a composition of its elements out of idempotents/generators,

  • 2.

    a way by which S is obtained from semilattices/cyclic groups acting on X, namely by means of bilateral semidirect products and quotients.

The point is to provide both (a) and (b) simultaneously while still being accountable for the resources used in terms of cardinalities. This approach is applied to the semigroup End(X, ⩽) of isotonic mappings of a linearly ordered set as well as the transition semigroups of automata that arise from certain varieties of formal languages. We discuss the semigroup varieties D, R, J, LJ1, and give a bilateral semidirect decomposition of the full transformation semigroup T(X) into End (X, ⩽) and the symmetric group on X.

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