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For every integer n with (n geqslant 6), we prove that the Boolean dimension of a poset consisting of all the subsets of ({1,dots ,n}) equipped with the inclusion relation is strictly less than n.

This paper is concerned with Riesz space structures on a lattice ordered abelian group, continuing a line of research conducted by the author and the collaborators Antonio Di Nola and Gaetano Vitale. First we prove a statement in a paper of Paul Conrad (given without proof) that every non-archimedean totally ordered abelian group has at least two Riesz space structures, if any. Then, as a main result, we prove that there is a non-archimedean lattice ordered abelian group with strong unit having only one Riesz space structure. This gives a solution to a problem posed in a paper of Conrad dating back to 1975. Then we combine these results and the categorial equivalence between lattice ordered abelian groups with strong unit and MV-algebras (due to Daniele Mundici) and the one between Riesz spaces with strong unit and Riesz MV-algebras (due to Di Nola and Ioana Leustean). By combining these tools, we prove that every non-semisimple totally ordered MV-algebra has at least two Riesz MV-algebra structures, if any, and that there is a non-semisimple MV-algebra with only one Riesz MV-algebra structure.

Given partially ordered sets (posets) ((P, le _P)) and ((P', le _{P'})), we say that (P') contains a copy of P if for some injective function (f:Prightarrow P') and for any (A, Bin P), (Ale _P B) if and only if (f(A)le _{P'} f(B)). For any posets P and Q, the poset Ramsey number R(P, Q) is the least positive integer N such that no matter how the elements of an N-dimensional Boolean lattice are colored in blue and red, there is either a copy of P with all blue elements or a copy of Q with all red elements. We focus on the poset Ramsey number (R(P, Q_n)) for a fixed poset P and an n-dimensional Boolean lattice (Q_n), as n grows large. It is known that (n+c_1(P) le R(P,Q_n) le c_2(P) n), for positive constants (c_1) and (c_2). However, there is no poset P known, for which (R(P, Q_n)> (1+epsilon )n), for (epsilon >0). This paper is devoted to a new method for finding upper bounds on (R(P, Q_n)) using a duality between copies of (Q_n) and sets of elements that cover them, referred to as blockers. We prove several properties of blockers and their direct relation to the Ramsey numbers. Using these properties we show that (R(mathcal {N},Q_n)=n+Theta (n/log n)), for a poset (mathcal {N}) with four elements A, B, C, and D, such that (A<C), (B<D), (B<C), and the remaining pairs of elements are incomparable.

Abstract

We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations (rho _1,dots ,rho _k) , is the downward closed set ({{,textrm{Av},}}(rho _1,dots ,rho _k)) consisting of all equivalence relations which do not contain any of (rho _1,dots ,rho _k) : (a) well-quasi-ordered, meaning that it contains no infinite antichains? and (b) atomic, meaning that it is not a union of two proper downward closed subsets, or, equivalently, that it satisfies the joint embedding property?

We prove that Higman’s lemma is strictly stronger for better quasi orders than for well quasi orders, within the framework of reverse mathematics. In fact, we show a stronger result: the infinite Ramsey theorem (for tuples of all lengths) is equivalent to the statement that any array ([mathbb N]^{n+1}rightarrow mathbb N^ntimes X) for a well order X and (nin mathbb N) is good, over the base theory (mathsf {RCA_0}).

A poset is a containment of paths in a tree (CPT) if it admits a representation by containment where each element of the poset is represented by a path in a tree and two elements are comparable in the poset if the corresponding paths are related by the inclusion relation. Recently Alcón, Gudiño and Gutierrez (Discrete Applied Math. 245, 139–147, 2018) introduced proper subclasses of CPT posets, namely dually-CPT, and strongly-CPT (or strong-CPT). A poset ({textbf{P}}) is dually-CPT, if ({textbf{P}}) and its dual ({textbf{P}}^{d}) both admit a CPT-representation. A poset ({textbf{P}}) is strongly-CPT, if ({textbf{P}}) and all the posets that share the same underlying comparability graph admit a CPT-representation. Where as the inclusion between dually-CPT and CPT was known to be strict. It was raised as an open question by Alcón, Gudiño and Gutierrez (Discrete Applied Math. 245, 139–147, 2018) whether strongly-CPT was a strict subclass of dually-CPT. We provide a proof that both classes actually coincide.

A generalized implication on a distributive lattice (varvec{A}) is a function between (varvec{A} times varvec{A}) to ideals of (varvec{A}) satisfying similar conditions to strict implication of weak Heyting algebras. Relative anihilators and quasi-modal operators are examples of generalized implication in distributive lattices. The aim of this paper is to study some classes of distributive lattices with a generalized implication. In particular, we prove that the class of Boolean algebras endowed with a quasi-modal operator is equivalent to the class of Boolean algebras with a generalized implication. This equivalence allow us to give another presentation of the class of quasi-monadic algebras and the class of compingent algebras defined by H. De Vries. We also introduce the notion of gi-sublattice and we characterize the simple and subdirectly irreducible distributive lattices with a generalized implication through topological duality.

Let (varvec{mathcal {A}(n,k)}) represent the collection of all (varvec{ntimes n}) zero-and-one matrices, with the sum of all rows and columns equalling (varvec{k}). This set can be ordered by an extension of the classical Bruhat order for permutations, seen as permutation matrices. The Bruhat order on (varvec{mathcal {A}(n,k)}) differs from the Bruhat order on permutations matrices not being, in general, graded, which results in some intriguing issues. In this paper, we focus on the maximum length of antichains in (varvec{mathcal {A}(n,k)}) with the Bruhat order. The crucial fact that allows us to obtain our main results is that two distinct matrices in (varvec{mathcal {A}(n,k)}) with an identical number of inversions cannot be compared using the Bruhat order. We construct sets of matrices in (varvec{mathcal {A}(n,k)}) so that each set consists of matrices with the same number of inversions. These sets are hence antichains in (varvec{mathcal {A}(n,k)}). We use these sets to deduce lower bounds for the maximum length of antichains in these partially ordered sets.