Globals of pseudovarieties of commutative semigroups: the finite basis problem, decidability and gaps (Q2716975)

The authors study pseudovarieties of commutative semigroups and their globals. Let \(\widehat\mathbb{N}\) be the profinite completion of the semiring of natural numbers \(\mathbb{N}\), \(\mathbb{P}=\mathbb{N}\setminus\{0\}\) and \(\widehat\mathbb{P}=\widehat\mathbb{N}\setminus\{0\}\). For \(m\in\mathbb{P}\cup\{0,\omega\}\), where \(\omega\) is the unique nonzero additive idempotent in \(\widehat\mathbb{N}\), and \(\pi\in\widehat\mathbb{P}\), let \(\mathbf{Com}_{m,\pi}\) denote the pseudovariety of all finite commutative semigroups satisfying the pseudoidentity \(x^{m+\pi}=x^m\). \(g\mathbf V\) denotes the global of the pseudovariety \(\mathbf V\).NEWLINENEWLINENEWLINETheorem 1: For any integer \(m\geq 2\), no pseudovariety \(\mathbf V\) between \(g\mathbf{Com}_{m,1}\) and \(g\mathbf{Com}_{m,\omega}\) has finite vertex rank. Theorem 2: For \(m\in\mathbb{N}\cup\{\omega\}\), \(\pi\in\widehat\mathbb{P}\), the following conditions are equivalent: (i) the pseudovariety of semigroupoids \(g\mathbf{Com}_{m,\pi}\) is decidable, (ii) the pseudovariety of semigroups \(\mathbf{Com}_{m,\pi}\) is decidable, (iii) it is decidable when a positive integer \(k\) divides \(\pi\). It remains open whether for every decidable pseudovariety \(\mathbf V\) of semigroups, \(g\mathbf V\) is decidable. An interval in a lattice is a big gap if it contains a continuum chain as well as a continuum antichain. Theorem 3: For all \(m\geq 2\), \(k\geq 1\), the interval of categorical pseudovarieties between \(g\mathbf{Com}_{m,k}\) and any of its successor skeleton points is a big gap.