An rj-normal lattice is a distributive lattice with 0 such that each prime ideal contains at most n minimal prime ideals. A relatively ra-normal lattice is a distributive lattice such that each bounded closed interval is an /¡.-normal lattice. The main results of this paper are: (1) a distributive lattice L with 0 is 71,-normal if and only if for any %n, x,,*•', X e L such that x Ax. = 0 for an...

The 2-category of constructively completely distributive lattices is shown to be bidual to a 2-category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2-category of ordered sets.

Bayesian probability theory is an inference calculus, which originates from a generalization of inclusion on the Boolean lattice of logical assertions to a degree of inclusion represented by a real number. Dual to this lattice is the distributive lattice of questions constructed from the ordered set of down-sets of assertions, which forms the foundation of the calculus of inquiry—a generalizati...

In this section we will talk about some simple algorithms for looking up items in a sorted list. Broadly speaking, an algorithm is a group of instructions that completely characterize the process of converting a set of inputs into desired outputs. To illustrate this concept, we shall solve a basic search problem. Let L = {l1, l2, . . . , lK} denote a set of ordered objects, where lk precedes lk...

A complete lattice L is constructively completely distributive CCD when the sup arrow from down closed subobjects of L to L has a left adjoint The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of CCD lattices and sup preserving arrows There is a restriction to order ideals and totally algebraic lattices Both biequivalences have left exact ver sions As appli...

England Abstract A stable homology theory is defined for completely distributive CSL algebras in terms of the point-neighbourhood homology of the partially ordered set of meet-irreducible elements of the invariant projection lattice. This specialises to the simplicial homology of the underlying simplicial complex in the case of a digraph algebra. These groups are computable and useful. In parti...

In this paper, we present equivalent characterizations of L-closure systems on lower bounded complete L-ordered sets and complete L-lattices, respectively. These results demonstrate the feasibility of notion of fuzzy closure system developed in our previous work (L.-K. Guo, G.-Q. Zhang, Q.-G. Li: Fuzzy closure systems on L-ordered sets. Math. Log. Quart. 57 (3) (2011), 281-291.). Also we develo...