Alpha-beta pruning is the algorithm of choice for searching game trees with position values taken from a totally ordered set, such as the set of real numbers. We generalize to game trees with position values taken from a partially ordered set, and prove necessary and sufficient conditions for alpha-beta pruning to be valid. Specifically, we show that shallow pruning is possible if and only if t...

Every lattice is isomorphic to a lattice whose elements are sets of sets and whose operations are intersection and the operation ∨∗ defined by A ∨∗ B = A ∪ B ∪ {Z : (∃X ∈ A)(∃Y ∈ B)X ∩ Y ⊆ Z}. This representation spells out precisely Birkhoff’s and Frink’s representation of arbitrary lattices, which is related to Stone’s set-theoretic representation of distributive lattices. (AMS Subject Classi...

Let FΓ be a finite dimensional path algebra of a quiver Γ over a field F. Let L and R denote the varieties of all left and right FΓ-modules respectively. We prove the equivalence of the following statements. • The subvariety lattice of L is a sublattice of the subquasivariety lattice of L. • The subquasivariety lattice of R is distributive. • Γ is an ordered forest.

We present a purely domain-theoretic model of Coquand and Huet’s Calculus of Construction [3], which is one of the most powerful type systems proposed in the literature. The well-formed expressions of its language are divided into three levels: Terms, Types, and Orders. Terms are the elements of Types, while the elements of Orders are called Operators. There is a special Order constant Type den...

The free distributive completion of a partial complete lattice is the complete lattice that it is freely generated by the partial complete latticèin the most distributive way'. This can be described as being a universal solution in the sense of universal algebra. Free distributive completions generalize the constructions of tensor products and of free completely distributive complete lattices o...

We show that for any tolerance R on U , the ordered sets of lower and upper rough approximations determined by R form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if R is induced by an irredundant covering of U , and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set RS of rough...

This paper focuses on the relationship between an $L$-subset and the system of level-elements induced by it, where the underlying lattice $L$ is a complete residuated lattice and the domain set of $L$-subset is an $L$-partially ordered set $(X,P)$. Firstly, we obtain the sufficient and necessary condition that an $L$-subset is represented by its system of level-elements. Then, a new representat...

Whaley’s Theorem on the existence of large proper sublattices of infinite lattices is extended to ordered sets and finite lattices. As a corollary it is shown that every finite lattice L with |L| ≥ 3 contains a proper sublattice S with |S| ≥ |L| 3 . It is also shown that that every finite modular lattice L with |L| ≥ 3 contains a proper sublattice S with |S| ≥ |L| 2 , and every finite distribut...

the starting point of this paper is given by priestley’s papers, where a theory of representation of distributive lattices is presented. the purpose of this paper is to develop a representation theory of fuzzy distributive lattices in the finite case. in this way, some results of priestley’s papers are extended. in the main theorem, we show that the category of finite fuzzy priestley space...

Let L be a finite distributive lattice, and let J(L) denote the set of all join-irreducible elements of L. Set j (L) = IJ(Z)l. For each a C J(L), let u(a) denote the number of elements in the prime filter {x C L: x >~a}. Our main theorem is Theorem 1. For any finite distributive lattice L, 4 "(a) ~>j(L)41q ,'2. aEJ(L) The base 4 here can most likely be replaced by a smaller number, but it canno...