Melissa Sugimoto
Characterization of Il-Semilattices
Overview: Boolean algebras (BAs) are algebras of subsets of sets with the operations union, intersection and complementation. We describe more general algebras called il-semilattices by proving that they are disjoint unions of BAs and can be represented by diagrams of partial functions. We hope to show how BAs can be glued to form all finite il-semilattices.Abstract: An involutive lattice-ordered semigroup (il-semigroup) is of the form (A, ≤, /\, \/, ∙, ~, -) such that (A, ≤, /\, \/) form a lattice, ∙ is an associative binary operation, and ~, - are an involutive pair. That is, for any x in A, -~x = x = ~-x, and for all x, y, z in A, x∙y ≤ z, x ≤ -(y∙~z), and y ≤ ~(-z∙x) are equivalent. Such a semigroup is idempotent if x∙x = x
and commutative if x∙y = y∙x. In this case (denoted by il-semilattices) the binary operation ∙ is a semilattice operation and the partial order it induces on the set A is called the multiplicative order. We prove that the multiplicative orders of all finite il-semilattices can be partitioned into Boolean algebras and represented by diagrams of partial functions, and we hope to show that conversely there exists a process of combining Boolean algebras such that every gluing of this form produces the multiplicative representation of an il-semilattice. A similar result has recently been shown in the related case of commutative idempotent involutive residuated lattices (P. Jipsen, O. Tuyt, and D. Valota’s preprint “Structural Characterization of Commutative Idempotent Involutive Residuated Lattices”), and it is our goal to prove these results in the il-semilattice case in order to give a full description of the structure of all finite il-semilattices.
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