2025/05/17 by Bonzio, Stefano, Gil-Férez, José, Jipsen, Peter +2
#FOS: Computer and information sciences #Logic in Computer Science (cs.LO)
paper · doi:10.48550/arxiv.2505.12024
A residuated semigroup is a structure ⟨ A,≤,⋅,\backslash,/ ⟩ where ⟨ A,≤ ⟩ is a poset and ⟨ A,⋅ ⟩ is a semigroup such that the residuation law x⋅ y≤ z\iff x≤ z/y\iff y≤ x \backslash z holds. An element p is positive if a≤ pa and a ≤ ap for all a. A residuated semigroup is called balanced if it satisfies the equation x \backslash x ≈ x / x and moreover each element of the form a \backslash a = a / a is positive, and it is called integrally closed if it satisfies the same equation and moreover each element of this form is a global identity. We show how a wide class of balanced residuated semigroups (so-called steady residuated semigroups) can be decomposed into integrally closed pieces, using a generalization of the classical Plonka sum construction. This generalization involves gluing a disjoint family of ordered algebras together using multiple families of maps, rather than a single family as in ordinary Plonka sums.