We introduce (ℓ-)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially (lattice-ordered) set. Bimonoids form an appropriate framework for the study general notion complementation, which subsumes both Boolean complements in bounded distributive lattices and multiplicative inverses monoids. The central question paper is whether how bimonoids can be embedded...

We synthesize a refutationally complete tableau calculus for propositional logic by a new method, using ordered resolution as a procedural framework at the meta level. Algebraically, the calculus solves the word problem for free distributive lattices. Goal-directedness of the construction is enforced by a ne-tuned well-founded ordering on terms, atoms and clauses. The synthesis is performed in ...

It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalisation of this fact and prove a converse of the generalisation. To be precise, we exhibit a one-to-one correspondence (up to isomorphism) between soft sheaf repr...

This paper proposes to study the lattice properties of two closed binary operations in the set of discrete fuzzy numbers. Using these operations to represent the meet and the join, we prove that the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers is a distributive lattice. Finally, we demonstrate that the subsets of discrete fuzzy numbers, which have the same...

Problems from Review of Set Theory Notes 3 (c) We prove that A× (B −C) = (A×B)− (A×C). We show that A× (B −C) ⊂ (A×B)−(A×C), and (A×B)−(A×C) ⊂ A×(B−C). For the first containment, let x ∈ A× (B − C). Then x = (a, b) where a ∈ A and b ∈ B − C. We certainly have x ∈ A × B, and since the second coordinate of x is not in C, we conclude that x / ∈ A× C. For the second containment, let x ∈ (A× B)− (A×...

We introduce a new method for ranked set sampling with multiple criteria. The method relaxes the restriction of selecting just one individual variable from each ranked set. Under the new method for ranking, units are ranked in sets based on linear extensions in partially order set theory with considering all variables simultaneously. Results willbe evaluated by a relatively extensive simulation...

We introduce Priestley rings of upsets (of a poset) and prove that inequivalent Priestley ring representations of a bounded distributive lattice L are in 1-1 correspondence with dense subspaces of the Priestley space of L. This generalizes a 1955 result of Bauer that inequivalent reduced field representations of a Boolean algebra B are in 1-1 correspondence with dense subspaces of the Stone spa...

If R is a commutative ring with identity and ≤ is defined by letting a ≤ b mean ab = a or a = b, then (R,≤) is a partially ordered ring. Necessary and sufficient conditions on R are given for (R,≤) to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings Zn of integers mod n for n ≥ 2. In particular, if R is reduced, then (R,≤) is a la...