The main theorem of this paper is a description of the class \({T_p}^q\) of finite semigroups that are triangularizable over the field of characteristic \(p\) and order \(q\), as an intersection of well-known pseudovarieties of semigroups and, thereby, as the class of finite semigroups satisfying a certain set of pseudoidentities. The term ‘triangularizable’ means that the given semigroup is embeddable in the semigroup of all upper triangular \(n\times n\) matrices over the given field, for some positive integer \(n\). Thus it is not a priori obvious that \({T_p}^q\) is a pseudovariety at all, since such classes must be closed under homomorphisms.
Motivation for this general study comes from classic language-theoretic results that relate the class of piecewise testable semigroups, the \(\mathcal J\)-trivial semigroups, and the semigroups of upper triangular Boolean matrices, and from extensions to mod-\(p\) analogues for each of these three classes. The main theorem states that a finite semigroup belongs to \({T_p}^q\) if and only if (1) each of its regular \(\mathcal D\)-classes is a subsemigroup, (2) each of its subgroups is an extension of a \(p\)-group by an Abelian group of exponent dividing \(q-1\), and (3) each subgroup of the subsemigroup generated by its idempotents is a \(p\)-group.