Tameness of Pseudovariety Joins Involving sf R{sf R}

In this paper, we establish several decidability results for pseudovariety joins of the form sf Vúsf W{sf V}vee{sf W} , where sf V{sf V} is a subpseudovariety of sf J{sf J} or the pseudovariety sf R{sf R} . Here, sf J{sf J} (resp. sf R{sf R} ) denotes the pseudovariety of all J{cal J} -trivial (resp. ?{cal R} -trivial) semigroups. In particular, we show that the pseudovariety sf Vúsf W{sf V}vee{sf W} is (completely) κ-tame when sf V{sf V} is a subpseudovariety of sf J{sf J} with decidable κ-word problem and sf W{sf W} is (completely) κ-tame. Moreover, if sf W{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 Rúsf W{sf R}vee{sf W} is also κ-tame. In particular the joins sf Rúsf Ab{sf R}vee{sf Ab} , sf Rúsf G{sf R}vee{sf G} , sf Rúsf OCR{sf R}vee{sf OCR} , and sf Rúsf CR{sf R}vee{sf CR} are decidable.