The Structure of Locally Integral Involutive Po-monoids and Semirings

Abstract

We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) A = (A, , •, ∼, −), and in particular every locally integral involutive semiring, decomposes in a unique way into a family {A p : p ∈ A + } of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms Φ = {ϕ pq : A p → A q : p q}, indexed over the positive cone (A + , ), so that the structure of A can be recovered as a glueing Φ A p of its integral components along Φ. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family of integral ipo-monoids {A p : p ∈ D}, indexed over a join-semilattice (D, ∨) along a family of monoid homomorphisms Φ is an ipo-semigroup.

Group Representation for Even and Odd Involutive Commutative Residuated Chains

“…We say that x ∈ Lu is u-invertible if there is y ∈ Lu such that xy = u. 7 It is tacitly understood that if a ∈ A ⊆ B then • a is the same for a ∈ A and for a ∈ B. Hence, for A ⊆ B,…”

“…It is called layer algebra decomposition. This idea was used recently for other classes of residuated lattices including finite commutative idempotent involutive residuated lattices in [9] and locally integral involutive po-monoids and semirings in [7]. In these classes the layer algebras are "nice".…”

“…This concept has been recently implemented to structurally describe various classes of residuated lattices, extending beyond its initial application. These encompass finite commutative, idempotent, and involutive residuated lattices, [13], finite involutive po-semilattices [14], and locally integral involutive po-monoids and semirings [3]. In these classes the layer algebras are "nice".…”

Logics of Variable Inclusion

“…[8]). Recently, the tools offered by Płonka sums have fruitfully been extended to the structural analysis of residuated structures, establishing a natural connection with substructural logics (see [21], [18]). In line with this trend, we will further extend the application of the method.…”