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.

For a monoid S, a (left) S-act is a non-empty set B together with a mapping S × B → B sending (s, b) to sb such that s(tb) = (st)b and 1b = b for all s, t ∈ S and b ∈ B. Right S-acts A can also be defined, and a tensor product A⊗S B (a set) can be defined that has the customary universal property with respect to balanced maps from A×B into arbitrary sets. Over the past three decades, an extensi...

Let C be a class of ordered algebras of a given fixed type τ . Associated with the type is a first order language Lτ , which must also contain a binary predicate to be interpreted by the ordering in members of C. One can then ask the question, when is the class C axiomatisable by sentences of Lτ? In this paper we will be considering axiomatisability problems for classes of left S-posets over a ...

in this paper the notion of rees short exact sequence for s-posets is introduced, and we investigate the conditions for which these sequences are left or right split. unlike the case for s-acts, being right split does not imply left split. furthermore, we present equivalent conditions of a right s-poset p for the functor hom(p;-) to be exact.

nets, useful topological tools, used to generalize certainconcepts that may only be general enough in the context of metricspaces. in this work we introduce this concept in an $s$-poset, aposet with an action of a posemigroup $s$ on it whichis a very useful structure in computer sciences and interestingfor mathematicians, and give the the concept of $s$-net. using $s$-nets and itsconvergency we...

Let C be a class of ordered algebras of a given fixed type τ . Associated with the type is a first order language Lτ , which must also contain a binary predicate to be interpreted by the ordering in members of C. One can then ask the question, when is the class C axiomatisable by sentences of Lτ? In this paper we will be considering axiomatisability problems for classes of left S-posets over a ...

Given a family of sets S, where the sets in S admit k ‘degrees of freedom’, we prove that not all (k + 1)-dimensional posets are containment posets of sets in S. Our results depend on the following enumerative result of independent interest: Let P (n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that logP (n, k) ∼ nk log n where k is fixed and n i...

‎In this paper some properties of weak regular injectivity for $S$-posets‎, ‎where $S$ is a pomonoid‎, ‎are studied‎. ‎The behaviour‎ ‎of different kinds of weak regular injectivity with products‎, ‎coproducts and direct sums is considered‎. ‎Also‎, ‎some‎ ‎characterizations of pomonoids over which all $S$-posets are of‎ ‎some kind of weakly regular injective are obtained‎. ‎Further‎, ‎we‎ ‎giv...