The notion of a $D$-poset was introduced in a connection withquantum mechanical models. In this paper, we introduce theconditional expectation of  random variables on theK^{o}pka's $D$-Poset and prove the basic properties ofconditional expectation on this  structure.

We provide a method to construct a type of orthomodular structure known as an orthoalgebra from the direct product decompositions of an object in a category that has finite products and whose ternary product diagrams give rise to certain pushouts. This generalizes a method to construct an orthomodular poset from the direct product decompositions of familiar mathematical structures such as non-e...

Uncertainty and fuzziness are involved in most of the real world problems. In present communication we have studied various advancements in fuzzy information theory and proposed a new measure of divergence between two fuzzy sets. Further, a measure of fuzzy directed divergence was developed based on the proposed divergence measure. Some properties of this fuzzy directed divergence measure were ...

Let Bn be the poset generated by the subsets of [n] with the inclusion as relation and let P be a nite poset. We want to embed P into Bn as many times as possible such that the subsets in di erent copies are incomparable. The maximum number of such embeddings is asymptotically determined for all nite posets P as 1 t(P ) ( n bn/2c ) , where t(P ) denotes the minimal size of the convex hull of a ...

Empty p0 p2 . . . pξ pξ+2 . . . Nonempty p1 p3 . . . pξ+1 pξ+3 . . . where pξ, ξ < α, is descending in P and Nonempty wins the game of length α if he can play α times. A poset P is called weakly α-game-closed if Player Nonempty has a winning strategy in the Banach-Mazur game of length α, where Nonempty is allowed to play at limit stages. P is called strongly α-game-closed if Player Nonempty has...

Building everything from scratch, we give another proof of Propp and Roby’s theorem saying that the average antichain size in any reverse operator orbit of the poset [m] × [n] is mn m+n . It is conceivable that our method should work for other situations. As a demonstration, we show that the average size of antichains in any reverse operator orbit of [m] ×Kn−1 equals 2mn m+2n−1 . Here Kn−1 is t...

the notion of a $d$-poset was introduced in a connection withquantum mechanical models. in this paper, we introduce theconditional expectation of  random variables on thek^{o}pka's $d$-poset and prove the basic properties ofconditional expectation on this  structure.

Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TCspanner) of G is a directed graph H = (V,EH) that has (1) the same transitive-closure as G and (2) diameter at most k. In some applications, the shortcut paths added to the graph in order to obtain small diameter can use Steiner vertices, that is, vertices not in the original graph G. The resulting spanne...

We introduce a new type of dcpo-completion of posets, called D-completion. For any poset P , the D-completion exists, and P and its Dcompletion have the isomorphic Scott closed set lattices. This completion is idempotent. A poset P is continuous (algebraic) if and only if its D-completion is continuous(algebraic). Using the D-completion, we construct the local dcpocompletion of posets, that rev...