We characterize the associative, idempotent, symmetric, and order-preserving binary operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In particular, we prove that the number of associative, idempotent, symmetric, and order-preserving operations on an n-element chain is the nth Catalan number.

For an effect algebra $A$, we examine the category of all morphisms from finite Boolean algebras into $A$. This category can be described as a category of elements of a presheaf $R(A)$ on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra $A$ can be characterized by properties of the category of elements of the presheaf $R(A)$. We prove that the tensor product of of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. As a consequence, the tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.

We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand-Tsetlin polytopes and cones, as well as Berenstein-Zelevinsky polytopes, all of which have appeared in the representation theory of semi-simple Lie algebras. The faces of these polyhedra correspond to certain partitions of the underlying poset and we give a combinatorial characterization of these partitions. We specify a class of marked posets that give rise to polyhedra with facets in correspondence to the covering relations of the poset. On the convex geometrical side, we describe the recession cone of the polyhedra, discuss products and give a Minkowski sum decomposition. We briefly discuss intersections with affine subspaces that have also appeared in representation theory and recently in the theory of finite Hilbert space frames.

Let F be a non-archimedean o-field and C be the field of generalized complex numbers over F. In this paper, we describe all the directed partial orders on C with 1 > 0 by using admissible semigroups of F +.

We investigate the minor order of functions, focusing on upper covers and common upper bounds of pairs of functions. We show that two functions of arities m and n have a common upper bound if and only if they have a common lower bound, and if a common upper bound exists, then there is one of arity m + n − 1. Moreover, we determine the possible essential arities of upper covers of functions.

We call a lattice crosscut-simplicial if the crosscut complex of every atomic interval is equal to the boundary of a simplex. Every interval of such a lattice is either contractible or homotopy equivalent to a sphere. Recently, Hersh and Meszaros introduced SB-labellings and proved that if a lattice has an SB-labelling then it is crosscut-simplicial. Some known examples of lattices with a natural SB-labelling include the join-distributive lattices, the weak order of a Coxeter group, and the Tamari lattice. Generalizing these three examples, we prove that every meet-semidistributive lattice is crosscut-simplicial, though we do not know whether all such lattices admit an SB-labelling. While not every crosscut-simplicial lattice is meet-semidistributive, we prove that these properties are equivalent for chamber posets of real hyperplane arrangements.

Motivated by work of Erdős, Milner and Rado, we investigate symmetric and asymmetric partition relations for linear orders without the axiom of choice. The relations state the existence of a subset in one of finitely many given order types that is homogeneous for a given colouring of the finite subsets of a fixed size of a linear order. We mainly study the linear orders 〈α2,<lex〉, where α is an infinite ordinal and <lex is the lexicographical order. We first obtain the consistency of several partition relations that are incompatible with the axiom of choice. For instance we derive partition relations for 〈ω2,<lex〉 from the property of Baire for all subsets of ω2 and show that the relation \(\langle ^{\kappa }{2}, <_{lex}\rangle \longrightarrow (\langle ^{\kappa }{2}, <_{lex}\rangle )^{2}_{2}\) is consistent for uncountable regular cardinals κ with κ<κ = κ. We then prove a series of negative partition relations with finite exponents for the linear orders 〈α2,<lex〉. We combine the positive and negative results to completely classify which of the partition relations \(\langle ^{\omega }{2}, <_{lex}\rangle \longrightarrow (\bigvee _{\nu <\lambda }K_{\nu },\bigvee _{\nu <\mu }M_{\nu })^{m}\) for linear orders Kν,Mν and m≤4 and 〈ω2,<lex〉→(K,M)n for linear orders K,M and natural numbers n are consistent.

For a partial order \(\mathbb {P}\) having infinite antichains by \(\mathfrak {a}(\mathbb {P})\) we denote the minimal cardinality of an infinite maximal antichain in \(\mathbb {P}\) and investigate how does this cardinal invariant of posets behave in finite products. In particular we show that \(\min \{ \mathfrak {a}(\mathbb {P}),\mathfrak {p} (\text {sq} \mathbb {P}) \} \leq \mathfrak {a} (\mathbb {P}^{n} ) \leq \mathfrak {a} (\mathbb {P})\), for all \(n\in \mathbb {N}\), where \(\mathfrak {p} (\text {sq} \mathbb {P})\) is the minimal size of a centered family without a lower bound in the separative quotient of the poset \(\mathbb {P}\), or \(\mathfrak {p} (\text {sq} \mathbb {P})=\infty \), if there is no such family. So we have \(\mathfrak {a} (\mathbb {P} \times \mathbb {P})=\mathfrak {a} (\mathbb {P})\) whenever \(\mathfrak {p} (\text {sq} \mathbb {P})\geq \mathfrak {a} (\mathbb {P})\) and we show that, in addition, this equality holds for all posets obtained from infinite Boolean algebras of size ≤o1 by removing zero, all reversed trees, all atomic posets and, in particular, for all posets of the form \(\langle \mathcal {C} ,\subset \rangle \), where \(\mathcal {C}\) is a family of nonempty closed sets in a compact T1-space containing all singletons. As a by-product we obtain the following combinatorial statement: If X is an infinite set and {Ai×Bi:i∈I} an infinite partition of the square X2, then at least one of the families {Ai :i∈I} and {Bi:i∈I} contains an infinite partition of X.