Tameness of pseudovariety joins involving R 1

In this paper, we establish several decidability results for pseudovariety joins of the form V ∨ W, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all J-trivial (resp. R-trivial) semigroups. In particular, we show that the pseudovariety V ∨ W is (completely) κ-tame when V is a subpseudovariety of J with decidable κ-word problem and W is (completely) κ-tame. Moreover, if W is a κ-tame pseudovariety which satisfies the pseudoidentity x1 · · ·xryzt = x1 · · ·xryztω, then we prove that R ∨ W is also κ-tame. In particular the joins R ∨ Ab, R ∨ G, R ∨ OCR, and R ∨ CR are decidable. 2000 Mathematics Subject Classification: 20M07 (primary); 20M05, 20M35, 68Q70 (secondary).