Unary algebras, semigroups and congruences on free semigroups

  • Authors:

  • Tatjana Petković;Miroslav Ćirić;Stojan Bogdanović

  • Affiliations:

  • Turku Centre for Computer Science and Department of Mathematics, University of Turku, Turku, FIN-20014, Finland;Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, Niš 18000, Serbia and Montenegro;Faculty of Economics, University of Niš, P.O. Box 121, Niš 18000, Serbia and Montenegro

  • Venue:
  • Theoretical Computer Science - Words, languages and combinatorics
  • Year:
  • 2004

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Abstract

In the triangle consisting of automata, languages and semigroups various correspondences of Eilenberg's type between languages and semigroups and between automata and languages are known, and it remains to establish similar connections between automata and semigroups. In this paper we consider a more general case by taking unary X-algebras instead of automata and we establish complete lattice isomorphisms between the lattices of σ-varieties of X-algebras, κ-varieties of semigroups and weakly invariant congruences on the free semigroup X+, where κ is the cardinality of X, between the lattices of generalized σ-varieties of X-algebras, generalized κ-varieties of semigroups and filters of the lattice of weakly invariant congruences on X+, and between the lattices of pseudo-σ-varieties of X-algebras and pseudo-κ-varieties of semigroups.