Mathematics
Presenter(s): Melissa Sugimoto
Advisor(s): Dr. Peter Jipsen
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 cidil-semigroups) 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 cidil-semigroups can be partitioned into Boolean algebras, and we hypothesize that conversely there exists a process of combining Boolean algebras such that every gluing of this form produces the multiplicative representation of a cidil-semigroup. 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-semigroup case in order to give a full description of the structure of finite commutative idempotent il-semigroups.