Canonical forms for free κ-semigroups
José Carlos Costa
Abstract
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's 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.
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