Hyperdecidability of pseudovarieties of orthogroups (Q2710431)

For a nonempty set \(A\) and a pseudovariety \(\mathcal W\) of semigroups, let \(\overline\Omega_A{\mathcal W}\) be the topological semigroup of all \(A\)-ary implicit operations on \(\mathcal W\). The pseudovariety of all finite semigroups is denoted by \(\mathcal S\), and \(p_{\mathcal W}\colon\overline\Omega_A{\mathcal S}\to\overline\Omega_A{\mathcal W}\) is the unique continuous homomorphism which fixes the elements of \(A\). For \(S\) a finite semigroup and \(\varphi\colon A\to S\) a mapping, let \(\overline\varphi\colon\overline\Omega_A{\mathcal S}\to S\) be the unique continuous homomorphism whose restriction to \(A\) is \(\varphi\). Let \(\Gamma\) be a directed multigraph and for every edge \(x\) of \(\Gamma\), let \(\alpha x\) and \(\omega x\) be the beginning vertex and the end vertex of \(x\), respectively. For a semigroup \(S\) and a labelling \(f\colon\Gamma\to S^1\) of \(\Gamma\) where the edges are labelled by elements of \(S\), \(f\) is said to be consistent if for every edge \(x\), \(f(\alpha x)f(x)=f(\omega x)\) in \(S\). The pseudovariety \(\mathcal W\) is said to be hyperdecidable if there exists an algorithm which decides whether for any labelling \(f\colon\Gamma\to S^1\), with \(\Gamma\) and \(S\) finite, there exists a labelling \(g\colon\Gamma\to(\overline\Omega_A{\mathcal S})^1\) such that \(\overline\varphi\circ g=f\) and such that \(h=p_{\mathcal W}\circ g\) is consistent.NEWLINENEWLINENEWLINEFor a pseudovariety \(\mathcal H\) of groups, let \({\mathcal B}\circm_{\text{CR}}{\mathcal H}\) be the pseudovariety which consists of the completely regular semigroups which divide finite semigroups \(S\) on which there exists a congruence \(\rho\) such that \(S/\rho\in{\mathcal H}\) and such that every \(\rho\)-class is a band. The main theorem of this paper states that if \(\mathcal H\) is hyperdecidable, then so is \({\mathcal B}\circm_{\text{CR}}{\mathcal H}\). As a result, the pseudovariety of all finite orthogroups is hyperdecidable.NEWLINENEWLINENEWLINEThe notion of a canonically reducible pseudovariety is a variant of the notion of a hyperdecidable pseudovariety and applies to pseudovarieties where the unary implicit operation \(\omega^{-1}\) of weak inversion is singled out for special attention. It is shown that the pseudovariety \({\mathcal B}\circm_{\text{CR}}{\mathcal H}\) is canonically reducible whenever the pseudovariety of groups \(\mathcal H\) is canonically reducible.