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On certain codes admitting inverse semigroups as syntactic monoids

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Abstract

The purpose of this paper is to investigate under what conditions an inverse semigroup M is isomorphic to the syntactic monoid M(A)* of afinite prefix code A over an alphabet X. We find a necessary condition for this to happen. It expresses a precise link between the group of units of M and the maximal subgroups of the 0-minimal ideal of M (Theorem 2.1). The condition is shown to be sufficient in case M is an ideal extension of a Brandt semigroup by a group (Corollary 2.3). We also introduce and study stable codes (products of subsets of the alphabet) and give structural properties of their syntactic monoids (Proposition 3.3 and Theorem 3.5). Most of our results inter-relate structural properties of certain semigroups and divisibility of integers attached to them. The terminology follows [1] and [3].

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Authors and Affiliations

  1. The Pennsylvania State University, 16802, University Park, Pa.

    Michael Keenan & Gerard Lallement

Authors
  1. Michael Keenan
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  2. Gerard Lallement
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TO ALFRED H. CLIFFORD on his 65th birthday on the 11th of July, 1973

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Keenan, M., Lallement, G. On certain codes admitting inverse semigroups as syntactic monoids. Semigroup Forum 8, 312–331 (1974). https://doi.org/10.1007/BF02194775

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  • DOI: https://doi.org/10.1007/BF02194775

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