A lot of combinatorial objects have algebra and coalgebra structures posets are important objects. In this paper, we construct on the vector space spanned by posets. Firstly, associativity unitary property, prove that with conjunction product is a graded algebra. Then definition free algebra, free. Finally, coassociativity counitary unshuffle coproduct coalgebra.

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.

In this paper, we introduce the concept of infinitely split Nash equilibrium in repeated games in which the profile sets are chain-complete posets. Then by using a fixed point theorem on posets in [8], we prove an existence theorem. As an application, we study the repeated extended Bertrant duopoly model of price competition.

A poset is said to be (2+ 2)-free if it does not contain an induced subposet that is isomorphic to 2+ 2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2+ 2)-free posets: P (t) = ∑ n≥0 ∏n i=1 ( 1− (1− t) ) . We extend this result by finding the generating function for (2+ 2)...

A retraction from a structure P to its substructure Q is a homomorphism from P onto Q that is the identity on Q. We present an algebraic condition which completely characterises all posets and all reflexive graphs Q with the following property: the class of all posets or reflexive graphs, respectively, that admit a retraction onto Q is first-order definable.

The first result presented in this paper is the closure under complementation of the class of languages of finite N-free posets recognized by branching automata. Relying on this, we propose a logic, named Presburger-MSO or P-MSO for short, precisely as expressive as branching automata. The P-MSO theory of the class of all finite N-free posets is decidable.

New properties that involve matchings, cutsets, or skipless chain partitions in graded posets are introduced and compared to familiar Sperner and chain partition properties. Related work is surveyed. We determine all possible combinations of these properties, with the exception of a long-standing open conjecture about L Y M posets, and provide a list of examples realizing these combinations.