Costa, José Carlos - Canonical forms for free κ-semigroups

dmtcs:1247 - Discrete Mathematics & Theoretical Computer Science, March 20, 2014, Vol. 16 no. 1
Canonical forms for free κ-semigroups

Authors: Costa, José Carlos

The implicit signature κ consists of the multiplication and the (ω-1)-power. We describe a procedure to transform each κ-term over a finite alphabet A into a certain canonical form and show that different canonical forms have different interpretations over some finite semigroup. The procedure of construction of the canonical forms, which is inspired in McCammond\textquoterights normal form algorithm for ω-terms interpreted over the pseudovariety A of all finite aperiodic semigroups, consists in applying elementary changes determined by an elementary set Σ of pseudoidentities. As an application, we deduce that the variety of κ-semigroups generated by the pseudovariety S of all finite semigroups is defined by the set Σ and that the free κ-semigroup generated by the alphabet A in that variety has decidable word problem. Furthermore, we show that each ω-term has a unique ω-term in canonical form with the same value over A. In particular, the canonical forms provide new, simpler, representatives for ω-terms interpreted over that pseudovariety.


Source : oai:HAL:hal-01179213v1
Volume: Vol. 16 no. 1
Section: Automata, Logic and Semantics
Published on: March 20, 2014
Submitted on: May 29, 2013
Keywords: Discrete Mathematics, Theoretical Computer Science,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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