Natanael Alpay
The Structure of Distributive Idempotent Weakly Conservative Lattice-ordered Magmas
Overview: In this project we studied the mathematical algebraic structures of distributive independent lattice ordered magmas. We proved some results and relationships related to these structures, which have many applications in graph theory, formal language, computer science, and abstract logic.Abstract: 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.
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