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Tameness of Pseudovariety Joins Involving ${\sf R}$

Authors:	Jorge Almeida José Carlos Costa Marc Zeitoun

Institution:	(1) Universidade do Porto, Portugal;(2) Universidade do Minho, Braga, Portugal;(3) Université Paris 7 and CNRS, France

Abstract:	In this paper, we establish several decidability results for pseudovariety joins of the form ${\sf V}\vee{\sf W}$ , where ${\sf V}$ is a subpseudovariety of ${\sf J}$ or the pseudovariety ${\sf R}$ . Here, ${\sf J}$ (resp. ${\sf R}$ ) denotes the pseudovariety of all ${\cal J}$ -trivial (resp. ${\cal R}$ -trivial) semigroups. In particular, we show that the pseudovariety ${\sf V}\vee{\sf W}$ is (completely) κ-tame when ${\sf V}$ is a subpseudovariety of ${\sf J}$ with decidable κ-word problem and ${\sf W}$ is (completely) κ-tame. Moreover, if ${\sf W}$ is a κ-tame pseudovariety which satisfies the pseudoidentity x₁ ⋯ x_ry^ω+1zt^ω = x₁ ⋯ x_ryzt^ω, then we prove that ${\sf R}\vee{\sf W}$ is also κ-tame. In particular the joins ${\sf R}\vee{\sf Ab}$ , ${\sf R}\vee{\sf G}$ , ${\sf R}\vee{\sf OCR}$ , and ${\sf R}\vee{\sf CR}$ are decidable. Partial support by FCT, through the Centro de Matemática da Universidade do Porto, is also gratefully acknowledged. Partial support by FCT, through the Centro de Matemática da Universidade do Minho, is also gratefully acknowledged.

Keywords:	2000 Mathematics Subject Classifications: 20M07 20M05 20M35 68Q70
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