We prove a weak version of the cross-product conjecture: $\textrm {F}(k+1,\ell ) \hskip .06cm \textrm {F}(k,\ell +1) \ge (\frac 12+\varepsilon ) \hskip .06cm \textrm {F}(k,\ell ) \hskip .06cm \textrm {F}(k+1,\ell +1)$, where $\textrm {F}(k,\ell )$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are k and $\ell $ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the generalized cross-product conjecture. The proofs use geometric inequalities for mixed volumes and combinatorics of words.

We develop universal algebra over an enriched category and relate it to finitary enriched monads over . Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.

Let $P$ be a finite $\text{N}$-free poset. We consider the hypergraph $H\left( P \right)$ whose vertices are the elements of $P$ and whose edges are the maximal intervals of $P$. We study the dual König property of $H\left( P \right)$ in two subclasses of $\text{N}$-free class.

We consider weighted sums over points of lattice polytopes, where the weight of a point v is the monomial qλ(v) for some linear form λ. We propose a q-analogue of the classical theory of Ehrhart series and Ehrhart polynomials, including Ehrhart reciprocity and involving evaluation at the q-integers. The main novelty is the proposal to consider q-Ehrhart polynomials. This general theory is then applied to the special case of order polytopes associated with partially ordered sets. Some more specific properties are described in the case of empty polytopes.

We investigate the partial orderings of the form P(X),⊂〉, where X is a relational structure and P(X) the set of the domains of its isomorphic substructures. A rough classification of countable binary structures corresponding to the forcing-related properties of the posets of their copies is obtained.

The complexity of infinite words is considered from the point of view of a transformation with a Mealy machine that is the simplest model of a finite automaton transducer. We are mostly interested in algebraic properties of the underlying partially ordered set. Results considered with the existence of supremum, infimum, antichains, chains and density aspects are investigated.

A Σ-labeled n-poset is an (at most) countable set, labeled in the set Σ, equipped with n partial orders. The collection of all Σ-labeled n-posets is naturally equipped with n binary product operations and nω-ary product operations. Moreover, the ω-ary product operations give rise to nω-power operations. We show that those Σ-labeled n-posets that can be generated from the singletons by the binary and ω-ary product operations form the free algebra on Σ in a variety axiomatizable by an infinite collection of simple equations. When n = 1, this variety coincides with the class of ω-semigroups of Perrin and Pin. Moreover, we show that those Σ-labeled n-posets that can be generated from the singletons by the binary product operations and the ω-power operations form the free algebra on Σ in a related variety that generalizes Wilke's algebras. We also give graph-theoretic characterizations of those n-posets contained in the above free algebras. Our results serve as a preliminary study to a development of a theory of higher dimensional automata and languages on infinitary associative structures.

Several “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices.