When does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular ^∗-semigroups and commutative involution semigroups.

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A normal form system (NFS) for representing Boolean functions is thought of as a set of stratified terms over a fixed set of connectives. For a fixed NFS A, the complexity of a Boolean function f with respect to A is the minimum of the sizes of terms in A that represent f. This induces a preordering of NFSs: an NFS A is polynomially as efficient as an NFS B if there is a polynomial P with nonnegative integer coefficients such that the complexity of any Boolean function f with respect to A is at most the value of P in the complexity of f with respect to B. In this paper we study monotonic NFSs, i.e., NFSs whose connectives are increasing or decreasing in each argument. We describe the monotonic NFSs that are optimal, i.e., that are minimal with respect to the latter preorder. We show that these minimal monotonic NFSs are all equivalent. Moreover, we address some natural questions, e.g.: does optimality depend on the arity of connectives? Does it depend on the number of connectives used? We show that optimal monotonic NFSs are exactly those that use a single connective or one connective and the negation. Finally, we show that optimality does not depend on the arity of the connectives.@en