Modular retractions of numerical semigroups

Let S be a numerical semigroup, let m be a nonzero element of S, and let a be a nonnegative integer. We denote ${\rm R}(S,a,m) = \{ s-as \bmod m \mid s \in S \}$ (where asmodm is the remainder of the division of as by m). In this paper we characterize the pairs (a,m) such that ${\rm R}(S,a,m)$ is a numerical semigroup. In this way, if we have a pair (a,m) with such characteristics, then we can reduce the problem of computing the genus of S=〈n ₁,…,n _p 〉 to computing the genus of a “smaller” numerical semigroup 〈n ₁?an ₁modm,…,n _p ?an _p modm〉. This reduction is also useful for estimating other important invariants of S such as the Frobenius number and the type.     

Modular translations of numerical semigroups

In this paper we introduce the concept of modular translation. With this tool, if we consider a certain numerical semigroup S, we build another one S′ whose principal invariants are given explicitly in terms of the invariants of S. Some results about irreducible numerical semigroups are also studied.     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

Schedule-induced posets

Schedules are a basic tool in the treatment of project networks and sequencing problems. For many questions, however, it is not the schedule itself but a partial order (poset) naturally induced on the set of activities that is really relied on. In fact, this correspondence of the usually uncountably many schedules to a finite system of posets was a key to many recent results in this field and constitutes the discrete character of the described problems, all the more so as additional properties of the respective schedules can very often be reflected in related properties of these induced posets.The present paper gives insights into the class of those posets that can be induced by schedules. An immediate observation is that schedule-induced posets are just the interval orders. More involved characterizations are concerned with the possible ‘local optimality’ of such structures. One such property is the existence of activity durations that will result in a project duration of the considered poset that is shorter (simultaneously) than the duration of all its extensions. A similar, related characterization for general regular cost functions (in particular tardiness cost) is also included, as are insights into ‘best’ activity durations with the mentioned properties.The paper closes with some remarks and numerical data on the considered class of posets as a whole.     

Quasicontinuous posets

Communicated by Michael Mislove     

Realizable posets

Certain p-local orders in n-dimensional division algebras over the rational numbers occur as endomorphism rings of torsion-free abelian groups of rank n if and only if an associated finite poset P has a strict faithful representation of dimension less than |P| over the field with p elements. In this note we obtain a simple characterization of those finite posets which do not admit such a representation.Research supported in part by NSF grant DMS-8802833.     

Multiplicative posets

A functionf from the posetP to the posetQ is a strict morphism if for allx, y P withx we havef(x). If there is such a strict morphism fromP toQ we writeP Q, otherwise we writeP Q. We say a posetM is multiplicative if for any posetsP, Q withP M andQ M we haveP ×Q M. (Here (p ₁,q ₁)<(p ₂,q ₂) if and only ifp ₁<p ₂ andq ₁<q ₂.) This paper proves that well-founded trees with height are multiplicative posets.This research was supported in part by NSERC Grant #69-1325.     

N-free posets as generalizations of series-parallel posets

N-Free posets have recently taken some importance and motivated many studies. This class of posets introduced by Grillet [8] and Heuchenne [11] are very related to another important class of posets, namely the series-parallel posets, introduced by Lawler [12] and studied by Valdes et al. [21]. This paper shows how N-free posets can be considered as generalizations of series-parallel posets, by giving a recursive construction of N-free posets. Furthermore we propose a linear time algorithm to recognize and decompose any N-free poset. This yields some very naturel problems, namely: which are the properties(such as linear time algorithm for some invariant) of series-parallel posets that are kept for N-free posets?     

Incidence posets of trees in posets of large dimension

In this paper, we investigate substructures of partially ordered sets which must be present whenever the dimension is large. We show that for eachn1, ifT is a tree onn vertices and ifP is any poset having dimension at least 4n ⁶, then eitherP or its dual contains the incidence poset ofT as a suborder.     

Counting finite posets and topologies

A refinement of an algorithm developed by Culberson and Rawlins yields the numbers of all partially ordered sets (posets) with n points and k antichains for n11 and all relevant integers k. Using these numbers in connection with certain formulae derived earlier by the first author, one can now compute the numbers of all quasiordered sets, posets, connected posets etc. with n points for n14. Using the well-known one-to-one correspondence between finite quasiordered sets and finite topological spaces, one obtains the numbers of finite topological spaces with n points and k open sets for n11 and all k, and then the numbers of all topologies on n14 points satisfying various degrees of separation and connectedness properties, respectively. The number of (connected) topologies on 14 points exceeds 10²³.     

III-Homomorphisms and III-congruences on posets

A new homomorphism between two partially ordered sets (the III-homomorphism) and a new congruence on a poset (the III-congruence) are introduced. Some properties of these homomorphisms and congruences and their relationship to the other known homomorphisms and congruences on posets are investigated. In contrast to total algebras, there are many different ways to introduce these notions. It is usually required that the respective notions should coincide with the usual definitions whenever lattices or semilattices are treated. The present paper presents an approach which in some sense completes the hierarchy of definitions so far used.     

On n-normal posets

A poset Q is called n-normal, if its every prime ideal contains at most n minimal prime ideals. In this paper, using the prime ideal theorem for finite ideal distributive posets, some properties and characterizations of n-normal posets are obtained.     

交C-连续偏序集

利用偏序集上的半拓扑结构,引入了交C-连续偏序集概念,探讨了交C-连续偏序集的性质、刻画及与C-连续偏序集、拟C-连续偏序集等之间的关系.主要结果有:(1)交C-连续的格一定是分配格;(2)有界完备偏序集(简记为bc-poset)L是交C-连续的当且仅当对任意x∈L及非空Scott闭集S,当∨S存在时有x∧∨S=∨{x∧s:s∈S};(3)完备格是完备Heyting代数当且仅当它是交连续且交C-连续的;(4)有界完备偏序集是C-连续的当且仅当它是交C-连续且拟C-连续的;(5)获得了反例说明分配的完备格可以不是交C-连续格,交C-连续格也可以不是交连续格.     

On multipartite posets

A poset P=(X,) is m-partite if X has a partition X=X₁X_m such that (1) each X_i forms an antichain in P, and (2) xy implies xX_i and yX_j where i<j. In this article we derive a tight asymptotic upper bound on the order dimension of m-partite posets in terms of m and their bipartite sub-posets in a constructive and elementary way.     

Quotient semigroups and extension semigroups

We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of C ^*-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.