Order-preserving Maps Research Articles

Semigroups of order preserving mappings on a finite chain: A new class of divisors

In this paper we aim to prove that every semigroup of the pseudovariety generated by all semigroups of partial, injective and order preserving transformations on a finite chain belongs to the pseudovariety generated by all semigroups of order preserving mappings on a finite chain.

A retract characterization of posets with the fixed-point property

It has been an open problem to characterize posets P with the property that every order-preserving map on P has a fixed point. We give a characterization of such posets in terms of their retracts.

Enriched 𝑃-Partitions

An (ordinary)PP-partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandard tableaux associated with Schur’sSS-functions. In this paper, we introduce and develop a theory of enrichedPP-partitions; like ordinaryPP-partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enrichedPP-partitions given here are the tableaux associated with Schur’sQQ-functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented.

Constructive order types, II

In this paper we continue the investigations begun in [2] of equivalence classes of (denumerable) well-orderings under one-one partial recursive, order-preserving maps (recursive isotonisms). The numbering and notation are continued from [2] except that we now use lower case Greek letters for classical ordinals and upper case Roman letters for sets and we slightly modify the definition of (see §IX.3).In section IX, after proving some lemmata which we use repeatedly in this paper, we establish the existence of Cantor Normal Forms for all co-ordinals less than a co-ordinal of the form WA (a fortiori for all co-ordinals less than a principal number for exponentiation).

Endomorphism regular Ockham algebras of finite Boolean type

AbstractIf (L; ƒ) is an Ockham algebra with dual space (X; g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and ƒ is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined.

Maximal sublattices of finite distributive lattices

Algebraic properties of lattices of quotients of finite posets are considered. Using the known duality between the category of all finite posets together with all order-preserving maps and the category of all finite distributive (0, 1)-lattices together with all (0, 1)-lattice homomorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of finite distributive (0, 1)-lattices are thereby obtained.

The complexity of the fixed point property

It is NP-complete to determine whether a given ordered set has a fixed point free order-preserving self-map. On the way to this result, we establish the NP-completeness of a related problem: Given ordered sets P and Q with t-tuples (p 1, ... , p t) and (q 1, ... , q t) from P and Q respectively, is there an order-preserving map f: P→Q satisfying f(p i)≥q i for each i=1, ... , t?

Logical quantizations of first-order structures

Many, though surely not all, mathematical structures can successfully be depicted by theories in (infinitary, many-sorted) first-order logic. The principal concern of this paper is to show systematically, following the lines of our previous papers, how to quantize a wide and important class of mathematical structures of first-order description, namely, the class of mathematical structures delineated by so-called limit theories. By way of example, not only every equational theory (e.g., the theory of groups and homomorphisms), but also the theory of partially ordered sets and order-preserving mappings and that of Banach spaces and contractive linear transformations are limit theories, so that they are susceptible of logical quantization.

Priestley duality for order-preserving maps into distributive lattices

The category of bounded distributive lattices with order-preserving maps is shown to be dually equivalent to the category of Priestley spaces with Priestley multirelations. The Priestley dual space of the ideal lattice Lσ of a bounded distributive lattice L is described in terms of the dual space of L. A variant of the Nachbin-Stone-Cech compactification is developed for bitopological and ordered spaces. Let X be a poset and Y an ordered space; XY denotes the poset of continuous order-preserving maps from Y to X with the discrete topology. The Priestley dual of LP is determined, where P is a poset and L a bounded distributive lattice.

The number of order-preserving maps between fences and crowns

We compute the number of order-preserving and -reversing maps between posets in the class of fences (zig-zags) and crowns (cycles).

Nilpotents in semigroups of partial one-to-one order-preserving mappings

Nilpotents in semigroups of partial one-to-one order-preserving mappings

Nilpotents in semigroups of partial order-preserving transformations

In this paper we extend the results of Garba [1] on IOn, the semigroup of all partial one-one order-preserving maps on Xn = {1,…, n}, to POn, the semigroup of all partial order-preserving maps on Xn, A description of the subsemigroup of POn generated by the set N of all its nilpotent elements is given. The set {α∈POn:lim α/≦r and |Xn /dom α|≧r} is shown to be contained in 〈N〉 if and only if r≦½n. The depth of 〈N〉, which is the unique k for which 〈N〉 = N ∪ N2 ∪…∪ Nk and 〈N〉 ≠ N ∪ N2 ∪…∪Nk−1 is shown to be equal to 3 for all n≧3. The rank of the subsemigroup {α∈POπ|imα|≦/n − 2 and α∈〈N〉} is shown to be equal to 6(n − 2), and its nilpotent rank to be equal to 7n−15.

Idempotent depth in semigroups of order-preserving mappings

We introduce algorithms for calculating minimum length factorisations of order-preserving mappings on a finite chain into products of idempotents, and into products of idempotents of defect one. The least upper bounds for these lengths are given.

Combinatorial results for semigroups of order-preserving mappings

Consider the finite set Xn = {1,2, …,n} ordered in the standard way. Let Tn denote the full transformation semigroup on Xn, that is, the semigroup of all mappings α: Xn→Xn under composition. We shall call α order-preserving if i ≤ j implies iα ≤ jα for i,j∈Xn, and α is decreasing if iα ≤ i for all i∈Xn. This paper investigates combinatorial properties of the semigroup O of all order-preserving mappings on Xn, and of its subsemigroup C which consists of all decreasing and order-preserving mappings.

Nonlinear Perron-Frobenius problem for order-preserving mappings, I

Nonlinear Perron-Frobenius problem for order-preserving mappings, I

On the automorphism conjecture for products of ordered sets

I. Rival and A. Rutkowski conjectured that the ratio of the number of automorphisms of an arbitrary poset to the number of order-preserving maps tends to zero as the size of the poset tends to infinity. We prove this hypothesis for direct products of arbitrary posets P=S1×⋯×Sn under the condition that maxi|Si|=0(√n/logn).

The number of order-preserving maps of fences and crowns

We perform an exact enumeration of the order-preserving maps of fences (zig-zags) and crowns (cycles). From this we derive asymptotic results.

The Dedekind-MacNeille completion as a reflector

We introduce a special type of order-preserving maps between quasiordered sets, the so-called cut-stable maps. These form the largest morphism class such that the corresponding category of quasiordered sets contains the category of complete lattices and complete homomorphisms as a full reflective subcategory, the reflector being given by the Dedekind-MacNeille completion (alias normal completion or completion by cuts). Suitable restriction of the object class leads to the category of separated quasiordered sets and its full reflective subcategory of completely distributive lattices. Similar reflections are obtained for continuous lattices, algebraic lattices, etc.

Dualities of Noetherian posets

In [10], J. D. Lawson obtained a duality between the category CP of continuous posets and CDL of completely distributive lattices. For these CP and CDL, the morphisms are special ones, namely, continuous order-preserving maps with the property that the inverse image of an open filter is a nonempty open filter, and frame-homomorphisms preserving coprimes, respectively. In this paper, we shall firstly give a duality between these two categories with continuous order-preserving maps and frame-homomorphisms, respectively. We then give some other dualities, which are obtained by the restrictions of this duality. Secondly, we shall deal with a duality between the category of Noetherian posets and p-complemented spatial frames (see definition in w For basic terminologies and concepts, one can refer to [6, 8] for lattices and frames, and to [3, 8] for categories. The author thanks the referee for many helpful suggestions.

Posets whose monoids of order-preserving maps are regular

The purpose of this paper is to determine all posetsP such that the monoid of all order-preserving maps ofP intoP is regular in the semigroup-theoretic sense.