| Abstract: | In this paper, we establish several decidability results for pseudovariety joins of the form
\sf Vú\sf 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 Vú\sf 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 ⋯ xryω+1ztω = x1 ⋯ xryztω, then we prove that
\sf Rú\sf W{\sf R}\vee{\sf W}
is also κ-tame. In particular the joins
\sf Rú\sf Ab{\sf R}\vee{\sf Ab}
,
\sf Rú\sf G{\sf R}\vee{\sf G}
,
\sf Rú\sf OCR{\sf R}\vee{\sf OCR}
, and
\sf Rú\sf CR{\sf R}\vee{\sf CR}
are decidable. |
