In our main result, we establish a formal connection between Lindström quantifiers with respect to regular languages and the double semidirect product of finite monoids with a distinguished set of generators. We use this correspondence to characterize the expressive power of Lindström quantifiers associated with a class of regular languages.

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities. As an example, an equational characterization of the pseudovariety corresponding to the class of regular languages in AC0 is given.

Motivated by the wavelength division multiplexing in all-optical networks, we consider the problem of finding an optimal (with respect to the least possible number of wavelengths) set of ƒ+1 internally node disjoint dipaths connecting all pairs of distinct nodes in the binary r-dimensional hypercube, where 0 ≤ ƒ < r. This system of dipaths constitutes a routing protocol that remains functional in the presence of up to ƒ faults (of nodes and/or links). The problem of constructing such protocols for general networks was mentioned in [1]. We compute precise values of ƒ-wise arc forwarding indexes and give (describe dipaths and color them) nearly optimal all-to-all ƒ-fault tolerant protocols for the hypercube network. Our results generalize corresponding results from [1, 4, 14].