Mathematics
Characterization of Commutative Idempotent il-Semigroups
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.
The structure of distributive idempotent weakly conservative lattice-ordered magmas
Presenter(s): Natanael Alpay
Advisor(s): Dr. Peter Jipsen
A lattice with 0 is an algebra (A,∧,∨,0) such that ∧,∨ are associative, commutative, absorptive (x ∨ (x ∧ y) = x = x ∧ (x ∨ y)) binary operations and x ∨ 0 = x. A lattice-ordered magma (l-magma for short) (A,∧,∨,0,·) is a lattice with 0 and a binary operation · such that x0 = 0 = 0x, x(y ∨ z) = xy ∨ xz and (x ∨ y)z = xz ∨ yz hold for all x, y, z ∈ A. A distributive idempotent l-magma (or dil-magma) is an l-magma A that satisfies x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and xx = x. Let J(A) be the set of completely join-irreducible elements of A, and define the property of weakly conservative as xy = x ∧ y or xy = x or xy = y or xy = x ∨ y for all x, y ∈ J(A). We show that every dually-algebraic weakly conservative dil-magma A is determined by two binary relations on the partially-ordered set J(A). In the case where the binary operation · is commutative and associative, and where the distributive lattice (A,∧,∨) is a complete and atomic Boolean algebra, we show that the structure of these algebras is determined by a preorder forest on the set of atoms of A. From these results we obtain efficient algorithms to construct all weakly conservative dil-magmas of size n and all Boolean commutative dil-semigroups of size 2^n.