Mathematics
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.