Modules whose submodule lattice is lower finite

A bounded poset [Formula: see text] is said to be lower finite if [Formula: see text] is infinite and for all [Formula: see text], there are but finitely many [Formula: see text] such that [Formula: see text] In this paper, we classify the modules [Formula: see text] over a commutative ring [Formula: see text] with identity with the property that the lattice [Formula: see text] of [Formula: see text]-submodules [Formula: see text] (under set-theoretic containment) is lower finite. Our results are summarized in Theorem 3.1 at the end of this note.

n-Permutability is not join-prime for n ≥ 5

Let [Formula: see text] be the variety generated by an order primal algebra of finite signature associated with a finite bounded poset [Formula: see text] that admits a near-unanimity operation. Let [Formula: see text] be a finite set of linear identities that does not interpret in [Formula: see text]. Let [Formula: see text] be the variety defined by [Formula: see text]. We prove that [Formula: see text] is [Formula: see text]-permutable for some [Formula: see text]. This implies that there is an [Formula: see text] such that [Formula: see text]-permutability is not join-prime in the lattice of interpretability types of varieties. In fact, it follows that [Formula: see text]-permutability where [Formula: see text] runs through the integers greater than 1 is not prime in the lattice of interpretability types of varieties. We strengthen this result by making [Formula: see text] and [Formula: see text] more special. We let [Formula: see text] be the 6-element bounded poset that is not a lattice and [Formula: see text] the variety defined by the set of majority identities for a ternary operational symbol [Formula: see text]. We prove in this case that [Formula: see text] is 5-permutable. This implies that [Formula: see text]-permutability is not join-prime in the lattice of interpretability types of varieties whenever [Formula: see text]. We also provide an example demonstrating that [Formula: see text] is not 4-permutable.

UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$ , for every $x,y$ . We show that the degrees of ceers under the equivalence relation generated by $\le$ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if $R\prime \le R$ , where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are ${\rm{\Sigma }}_3^0$ -complete (the former answering an open question of Gao and Gerdes).

The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck Paths

International audience In the first part of this article we present a realization of the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$ in terms of $m$-tuples of Dyck paths of height $n$, equipped with componentwise rotation order. For that, we define the $m$-cover poset $\mathcal{P}^{\langle m \rangle}$ of an arbitrary bounded poset $\mathcal{P}$, and show that the smallest lattice completion of the $m$-cover poset of the Tamari lattice $\mathcal{T}_n$ is isomorphic to the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$. A crucial tool for the proof of this isomorphism is a decomposition of $m$-Dyck paths into $m$-tuples of classical Dyck paths, which we call the strip-decomposition. Subsequently, we characterize the cases where the $m$-cover poset of an arbitrary poset is a lattice. Finally, we show that the $m$-cover poset of the Cambrian lattice of the dihedral group is a trim lattice with cardinality equal to the generalized Fuss-Catalan number of the dihedral group. Dans la première partie de cet article nous présentons une réalisation du treillis $m$ -Tamari $\mathcal{T}_n^{(m)}$ à l’aide de $m$-uplets de chemins de Dyck de hauteur $n$, équipés de l’ordre de rotation composante par composante. Pour cela, nous définissons le poset de $m$-couverture $\mathcal{P}^{\langle m \rangle}$ d’un poset borné quelconque $\mathcal{P}$, et montrons que la plus petite complétion en treillis du poset de $m$-couverture du treillis de Tamari $\mathcal{T}_n$ est isomorphe au treillis $m$-Tamari $\mathcal{T}_n^{(m)}$. Unoutil crucial pour la preuve de cet isomorphisme est une décomposition des chemins $m$-Dyck en $m$-uplets de chemins de Dyck usuels, que nous appelons la décomposition en bandes. Par la suite, nous caractérisons les cas où le poset de $m$-couverture d’un poset donné est un treillis. Enfin nous montrons que le poset de $m$-couverture du treillis Cambrien du groupe diédral est un treillis svelte de cardinalité le nombre généralisé de Fuss-Catalan du groupe diédral.

Extended blocker, deletion, and contraction maps on antichains

Families of maps on the lattice of all antichains of a finite bounded poset that extend the blocker, deletion, and contraction maps on clutters are considered. Influence of the parameters of the maps is investigated. Order-theoretic extensions of some principal relations for the set-theoretic blocker, deletion, and contraction maps on clutters are presented.

On blockers in bounded posets

Antichains of a finite bounded poset are assigned antichains playing a role analogous to that played by blockers in the Boolean lattice of all subsets of a finite set. Some properties of lattices of generalized blockers are discussed.