Totally Ordered Monoids Based on Triangular Norms

A totally ordered monoid, or tomonoid for short, is a monoid together with a translation-invariant (i.e., compatible) total order. We consider in this paper tomonoids fulfilling the following conditions: the multiplication is commutative; the monoidal identity is the top element; all nonempty suprema exist; and the multiplication distributes over arbitrary suprema. The real unit interval endowed with its natural order and a left-continuous t-norm is our motivating example. A t-norm is a binary operation used in fuzzy logic for the interpretation of the conjunction.

Given a tomonoid of the indicated type, we consider the chain of its quotients induced by filters. The intention is to understand the tomonoid as the result of a linear construction process, leading from the coarsest quotient, which is the one-element tomonoid, up to the finest quotient, which is the tomonoid itself. Consecutive elements of this chain correspond to extensions by Archimedean tomonoids. If in this case the congruence classes are order-isomorphic to real intervals, a systematic specification turns out to be possible.

In order to deal with tomonoids and their quotients in an effective and transparent way, we follow an approach with a geometrical flavor: we work with transformation monoids, using the Cayley representation theorem. Our main results are formulated in this framework. Finally, a number of examples from the area of t-norms are included for illustration.     

Ideals of Finitely Generated Commutative Monoids

Abstract. We extend the concept of presentation of finitely generated commutative monoids to ideals of finitely generated commutative monoids and give algorithms to obtain information about an ideal from a given presentation.     

Atomic Commutative Monoids and Their Elasticity

We present some characterizations of properties related to factorization problems on finitely generated commutative monoids. We also give a series of algorithms for studying these problems from a presentation of a given commutative monoid.     

Zigzag Theorem for Partially Ordered Monoids

While investigating the connection of absolute flatness of partially ordered monoids (briefly pomonoids) with their amalgamation, the author recently posed the question as to whether it was possible to formulate some ordered analogue of Isbell's zigzag theorem. Proving such a theorem is the primary aim of this article.     

On Morita Equivalence of Partially Ordered Monoids

We show that there is one-to-one correspondence between certain algebraically and categorically defined subobjects, congruences and admissible preorders of S-posets. Using preservation properties of Pos-equivalence functors between Pos-categories we deduce that if S and T are Morita equivalent partially ordered monoids and F:Pos _S →Pos _T is a Pos-equivalence functor then an S-poset A _S and the T-poset F(A _S ) have isomorphic lattices of (regular, downwards closed) subobjects, congruences and admissible preorders. We also prove that if A _S has some flatness property then F(A _S ) has the same property.     

A Note on the n-Generator Property for Commutative Monoids

Let M be a cancellative, commutative monoid with integral closure . Borrowing from ring theory, we say that M has the n-generator property iff every finitely generated ideal of M can be generated by n elements, and we say M has rank n iff every ideal of M can be generated by n elements. We investigate the integral closure of such monoids. We show, in particular, that if M has the n-generator property, then is a valuation monoid, and if M has rank n, then is a principal ideal monoid.     

Bitopologies on Totally Ordered Sets

We study the category of ray bispaces, that is, the category whose objects are totally ordered sets with two topologies, each having a subbase of rays and so that the resulting bitopological space is pairwise weakly symmetric, and whose morphisms are the pairwise continuous functions. In contrast with the purely topological results of [5], we show that, (1) such spaces are utterly normal and hence monotonically normal (in the sense of [6]), and (2) (Intermediate Value Theorem) the pairwise continuous image of a pairwise connected bitopological space in a selective ray bispace is an interval. We also obtain conditions for the equality of the de Groot dual (see [4]) and the ray dual (see [5]) of a ray topology and show that a selective ray topology is compact if and only if it is skew compact.     

关于交换序半群的一个注记

给出了正则交换序半群的与素理想结构有关的若干性质，还给出了为有限个主理想的并的交换序半群类中的诺特性，阿基米德性，正则性以及有限生成性之间的一些关系。     

Ordered Compactifications of Products of Two Totally Ordered Spaces

We describe the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y where X and Y are certain totally ordered topological spaces, and where _o Z denotes the Stone–ech ordered- or Nachbin-compactification of Z. These basic cases are used to illustrate techniques for describing the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y for arbitrary totally ordered topological spaces X and Y. Such products X × Y provide many counterexamples in the theory of ordered compactifications.     

The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids

Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then the catenary degree of S (denoted c(S)) and the tame degree of S (denoted t(S)) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we describe methods to compute both c(S) and t(S) when M is a finitely generated commutative cancellative monoid.     

Equational Classes of Totally Ordered Modal Lattices

A modal lattice is a bounded distributive lattice endowed with a unary operator which preserves the join-operation and the smallest element. In this paper we consider the variety CH of modal lattices that is generated by the totally ordered modal lattices and we characterize the lattice of subvarieties of CH. We also give an equational basis for each subvariety of CH.     

Group Codes Defined on Totally Ordered Sets

ＧｒｏｕｐＣｏｄｅｓＤｅｆｉｎｅｄｏｎＴｏｔａｌｌｙＯｒｄｅｒｅｄＳｅｔｓＷａｎＺｈｅｘｉａｎ（万哲先）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ，１０００８０ａｎｄＤｅｐａｒｔｍｅｎｔｏｆＩｎ...     

Lattice-Ordered Matrix Rings Over Totally Ordered Rings

For an n ×n matrix algebra over a totally ordered integral domain, necessary and sufficient conditions are derived such that the entrywise lattice order on it is the only lattice order (up to an isomorphism) to make it into a lattice-ordered algebra in which the identity matrix is positive. The conditions are then applied to particular integral domains. In the second part of the paper we consider n ×n matrix rings containing a positive n-cycle over totally ordered rings. Finally a characterization of lattice-ordered matrix ring with the entrywise lattice order is given.     

Order-Compactifications of Totally Ordered Spaces: Revisited

Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13:56–65, 1975) and by Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorčuk (Soviet Math Dokl 7:1011–1014, 1966; Sib Math J 10:124–132, 1969) and Kaufman (Colloq Math 17:35–39, 1967). In this note we give a simple characterization of the topology of a totally ordered space, as well as give a new simplified proof of the main results of Blatter (J Approx Theory 13:56–65, 1975) and Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Our main tool will be an order-topological modification of the Dedekind-MacNeille completion. In addition, for a zero-dimensional totally ordered space X, we determine which order-compactifications of X are Priestley order-compactifications.     

可换序半群的分式扩张

In this paper we extend the construction of the field of rational numbers from the ring of integers to an arbitrary commutative ordered semigroup.We first construct a fractional ordered semigroup and a homomorphism ψs:R→S-1R.Secondly,we characterize the commutative ordered semigroup so constructed by a universal mapping property.     

The Lattice of Ordered Compactifications of a Direct Sum of Totally Ordered Spaces

The lattice of ordered compactifications of a topological sum of a finite number of totally ordered spaces is investigated. This investigation proceeds by decomposing the lattice into equivalence classes determined by the identification of essential pairs of singularities. This lattice of equivalence classes is isomorphic to a power set lattice. Each of these equivalence classes is further decomposed into equivalence classes determined by admissible partially ordered partitions of the ordered Stone–ech remainder. The lattice structure within each equivalence class is determined using an algorithm based on the incidence matrix of the partially ordered partition. As examples, the ordered compactification lattices for the spaces [0,1)[0,1),[0,1)[0,1)[0,1),RR, and R/{0}R/{0} are determined.     

交换偏序半群的偏序扩张

设(S,·,≤)为偏序可换半群,本文给出将S的偏序≤扩张为满足一定条件的偏序≤^{*的充要条件.特别地,如果(S,·,≤)为可消偏序可换幺半群,本文给出将S的偏序≤扩张为可消偏序≤*且S的每个元素在≤*下均在正锥中的充要条件.本文还给出将偏序可换幺半群S的偏序≤扩张为≤*且使得S的有限元素子集在≤*下是一条链的充要条件.}     

On Totally Ordered Rings Whose Positive Cones are Finitely Generated

This paper is mainly dealt with the structure of totally ordered rings (t.o. rings) whose positive cones are finitely generated as multiplicative right ideals.AMS Subject Classification (1991): 16W80, 13J25