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In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain. [ABSTRACT FROM AUTHOR]

We give a presentation for the monoid I O n of all order-preserving transformations of an n-chain whose ranges are intervals. We also consider the submonoid I O n - of I O n consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation. [ABSTRACT FROM AUTHOR]

AbstractFor n∈Z+ , let Xn={1,…,n} be a finite chain and Tn be the full transformation semigroup on Xn . If OCTn is the set of all order-preserving and full contraction mappings on Xn , then it is well-known that OCTn is a submonoid of Tn . In this paper we give a monoid presentation of OCTn . [ABSTRACT FROM AUTHOR]

For a Coxeter system (W, S) and a subset J ⊆ S, fix u ∈ WJ and ξ ∈ WJ, let λ(−, u) : WJ → W and λ(ξ, −) : WJ → W be the mappings defined by λ(x, u) = xu and λ(ξ, y) = ξy, respectively. We show that λ(−, u) is left weak Bruhat order preserving for all u ∈ WJ, and that λ(ξ,−) is left weak Bruhat order-preserving if and only if it is a convex embedding, if and only if `(y) = `(ξyξ−1 ) for all y ∈ WJ. For the Coxeter groups of types A and B, we explicitly describe the ξ that satisfies the previous equivalent conditions in a purely combinatorial manner. [ABSTRACT FROM AUTHOR]

We study a submonoid of the well studied monoid P O I n of all order-preserving partial injections on an n-element chain. The set I O F n par of all partial transformations in P O I n which are fence-preserving as well as parity-preserving forms a submonoid of P O I n . We describe Green's relations and ideals of I O F n par . For each ideal of I O F n par , we characterize the maximal subsemigroups. We observe that there are three different types of maximal subsemigroups. [ABSTRACT FROM AUTHOR]

Let O P D n be the semigroup consisting of all orientation-preserving and order-decreasing full transformations on the finite chain X n = { 1 < ⋯ < n } , and let N (O P D n) be the subsemigroup consisting of all nilpotent elements of O P D n . Moreover, for 1 ≤ r ≤ n - 1 , let O P D (n , r) = { α ∈ O P D n : | im (α) | ≤ r } , and let N (O P D (n , r)) be the subsemigroup consisting of all nilpotent elements of O P D (n , r) . In this paper, we compute the cardinalities of O P D n , N (O P D n) , O P D (n , r) and N (O P D (n , r)) , and find their ranks. Moreover, for each idempotent ξ in O P D n , we show that O P D n (ξ) = { α ∈ O P D n α m = ξ for some m ∈ N } is the maximal nilpotent subsemigroup of O P D n with zero ξ , and we find its cardinality and rank. [ABSTRACT FROM AUTHOR]

In this paper, we study the cyclic inverse monoid ℐ n on a set Ω n with n elements, i.e. the inverse submonoid of the symmetric inverse monoid on Ω n consisting of all restrictions of the elements of a cyclic subgroup of order n acting cyclically on Ω n . We show that ℐ n has rank 2 (for n ≥ 2) and n 2 n − n + 1 elements. Moreover, we give presentations of ℐ n on n + 1 generators and 1 2 (n 2 + 3 n + 4) relations and on 2 generators and 1 2 (n 2 − n + 6) relations. We also consider the remarkable inverse submonoid ℐ n of ℐ n constituted by all its order-preserving transformations. We show that ℐ n has rank n and 3 ⋅ 2 n − 2 n − 1 elements. Furthermore, we exhibit presentations of ℐ n on n + 2 generators and 1 2 (n 2 + 3 n + 8) relations and on n generators and 1 2 (n 2 + 3 n) relations. [ABSTRACT FROM AUTHOR]

The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner–Preston Theorem. Let n be a positive natural number and PFI n be the semigroup of all fence-preserving partial one-to-one maps of { 1 , ... , n } into itself with respect to composition of maps and the fence 1 ≺ 2 ≻ 3 ≺ ⋯   n. There is considered the inverse semigroup IOF n par of all α ∈ PFI n such that α is regular in PFI n , order-preserving with respect to the order 1 < 2 < ⋯ < n and parity-preserving. According to the main result of the paper, it is 3 n − 6 the least of the cardinalities of the generating sets of IOF n par for 4 ≤ n. There is determined a concrete representation of a generating set of minimal size. [ABSTRACT FROM AUTHOR]

We study isotone self-maps of complete lattices and their fixed point sets, which are complete lattices contained as suborders, but not necessarily as subsemilattices. We develop a representation of such maps by means of relations and show how to navigate their fixed point lattices using a modification of the standard Next closure algorithm. Our approach is inspired by early work of Shmuely [8] and Crapo [1]. We improve and substantially extend our earlier publication [4]. [ABSTRACT FROM AUTHOR]

Let X = { 1 , ... , n } be a finite chain and let ℐ (X) be the semigroup of all injective order-preserving partial transformations on X. For any nonempty subset Y of X , let ℐ (X , Y) be the subsemigroup of ℐ (X) of all transformations with range contained in Y. In this paper, we characterize Green's relations on ℐ (X , Y) , and show that the semigroup ℐ (X , Y) is left abundant but not right abundant when Y is a proper subset of X. Moreover, the cardinality and the rank of the semigroup ℐ (X , Y) is determined. [ABSTRACT FROM AUTHOR]

Copyright of Electric Power Automation Equipment / Dianli Zidonghua Shebei is the property of Electric Power Automation Equipment Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let X be a chain and let O (X) be the monoid of all full order-preserving transformations of X. Given a nonempty subset Y of X, we denote by O (X , Y) the subsemigroup of O (X) of all full order-preserving transformations with range contained in Y and by O (Y) the monoid of all order-preserving transformations of Y. In 2015, Sommanee and Sanwong investigated the subsemigroup O F (X , Y) = { α ∈ O (X , Y) ∣ im (α) = Y α } of O (X , Y) . We characterize the connections between the maximal (regular) subsemibands of O F (X , Y) and the maximal (regular) subsemibands of O r = O (Y) \ { 1 Y } . Moreover, we characterize the structure of the idempotent generating sets of O F (X , Y) when Y is a finite subset of X. As applications, we compute the number of distinct minimal idempotent generating sets of O F (X , Y) . We also determine the maximal subsemibands as well as the maximal regular subsemibands of O F (X , Y) . [ABSTRACT FROM AUTHOR]

Let I n be the symmetric inverse semigroup on X n = { 1 , 2 , ⋯ , n } , and let O D P n be the subsemigroup of I n consisting of all order-preserving partial isometries of X n . Let Y = { 1 , 2 , ⋯ , r } be a non-empty subset of X n . Define O D P I n , r = { α ∈ O D P n : im α ⊆ Y }. In this paper, we give a necessary and sufficient condition for O D P I n , r to be an inverse semigroup and characterize its Green relations. Moreover, the cardinality and the rank of O D P I n , r are investigated. [ABSTRACT FROM AUTHOR]

Surveillance cameras have been widely used in urban environments and are increasingly used in rural ones. Such cameras have mostly been used for security, but they can be applied to the problem of furnishing fine-grained measurements and predictions of precipitation intensity. In this study, we formulated a stacked order-preserving (OP) learning framework to train a network using time-series data. We constructed an OP module, which uses a three-dimensional (3D) convolution operation to extract features with spatial and temporal information and that are associated with ConvLSTM; this feature extraction is used to learn the short-term and OP time-series relationships between features. Furthermore, the OP modules are stacked to form a stacked OP network (SOPNet), which strengthens the relationship between features in long-term time-series image sequences. This SOPNet can be use to obtain fine-grained measurements and predictions of precipitation intensity from images captured by outdoor surveillance cameras. Our main contributions are threefold. First, the SOPNet strengthens the short-term and long-term time-series relationship between features. Second, the SOPNet simultaneously examines spatial and temporal information to measure and predict precipitation intensity. Third, we constructed a precipitation intensity database based on optical images captured by outdoor surveillance cameras. We experimentally evaluated our proposed architecture using our self-collected data set. We found that SOPNet yields better performance and greater accuracy relative to its well-known state-of-the-art counterparts with respect to various metrics.

The purpose of this paper is the study of congruences on semigroups of transformations on a countably infinite fence. We consider the monoid O ℕ of all full transformations on the set ℕ of all natural numbers preserving the zig-zag order on ℕ , as well as the monoid E (O ℕ ∗) of all idempotent transformations in O ℕ additionally preserving the usual linear order on ℕ. We show that there are uncountably many congruences on O ℕ and determine seven maximal congruences on O ℕ , which are all the maximal congruences containing a particular congruence on O ℕ. Moreover, we characterize all congruences on the monoid of all transformations in E (O ℕ ∗) with infinite rank. For the semigroup of all transformations in E (O ℕ ∗) with finite rank, we determine the Rees congruences. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of an orientation-preserving transformation on an arbitrary chain, as a natural extension for infinite chains of the well known concept for finite chains introduced in 1998 by McAlister and, independently, in 1999 by Catarino and Higgins. We consider the monoid POP (X) (X) of all orientation-preserving partial transformations on a finite or infinite chain X and its submonoids and POPI (X) of all orientation-preserving full transformations and of all orientation-preserving partial permutations on X, respectively. The monoid OPPO (X) of all order-preserving partial transformations on X and its injective counterpart POI (X) are also considered. We study the regularity and give descriptions of the Green's relations of the monoids POP (X) , PO (X) , OP (X) , POPI (X) , and POI (X). [ABSTRACT FROM AUTHOR]

Abstract This paper addresses the stability problem of a class of switched positive nonlinear systems (SPNSs). Both continuous-time systems and discrete-time systems are studied. By applying the minimum dwell-time (MDT) approach, we design time-dependent switching rules under which the continuous-time SPNSs is asymptotically stable. For the corresponding discrete-time case, a sufficient condition is given for exponential stability of SPNSs. In addition, the MDT switching signals are designed via analyzing the weighted l∞ $l_{\infty}$ norm for the considered systems. Finally, a numerical example is provided to illustrate the effectiveness of our result.

A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup FN of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that FN has no minimal generating sets and present two particular infinite decreasing chains of generating sets of FN. [ABSTRACT FROM AUTHOR]

It is well known that the semigroup T(X) of all transformations on a set X is regular, but its subsemigroups do not need to be. Consider an ordered set (X ; ≤) whose order forms a path with alternating orientation. The semigroup OT (X) of all order-preserving transformations on (X ; ≤) is a subsemigroup of T(X). Some characterizations of regular semigroups OT (X) were given. For a non-empty subset S of X, we define the set OT S (X) = { α ∈ OT (X) ∣ ran α = S }. In this paper, we show that in general, OT S (X) is not a subsemigroup of T(X). A necessary and sufficient condition for OT S (X) to be a regular subsemigroup of T(X) is given. Some regular subsemigroups of T(X) that are contained in OT S (X) are studied. We investigate the left (right) regularity of OT (X) . We obtain that left regular and right regular elements in OT (X) are identical. A characterization of left (right) elements in OT (X) is given. We classify all left (right) regular semigroups OT (X) . [ABSTRACT FROM AUTHOR]

We characterize the monoids of endomorphisms of the semigroup of all order-preserving partial transformations and of the semigroup of all order-preserving partial permutations of a finite chain. [ABSTRACT FROM AUTHOR]

A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup TF n of all order-preserving transformations on an n-element zig-zag-ordered set. We determine the rank of TF n and provide a minimal generating set for TF n . Moreover, a formula for the number of idempotents in TF n is given. [ABSTRACT FROM AUTHOR]

For n ∈ N , let O n be the semigroup of all order-preserving transformations on the finite chain X n = { 1 , ... , n } , under its natural order. For any non-empty subset A of X n , let O n (A) and O n + (A) be the subsemigroups of all order-preserving and A-decreasing, and of all order-preserving and A-increasing transformations on X n , respectively. In this paper we obtain formulae for the number of elements and for the number of idempotents in O n (A) . Moreover, we show that O n (A) contains a zero element if and only if 1 ∈ A , and then we obtain the number of nilpotents in O n (A) when 1 ∈ A . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the irreducible generating sets of orthogonal and symplectic rook monoids. We find that these irreducible generating sets are closely related to those of the unit groups of the monoids. We study and determine irreducible generating sets of order-preserving orthogonal rook monoids and order-preserving symplectic rook monoids. We then investigate nonirreducible generating sets of the singular parts of the orthogonal and symplectic rook monoids. [ABSTRACT FROM AUTHOR]

In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of quantum states was used to establish the existence of a fixed quantum state associated to an order-preserving quantum operation. The method we presented is an alternative in showing the existence of a fixed quantum state associated to quantum operations. Our method does not capitalise on the commutativity of the quantum effects with the fixed quantum state(s) (Luders’s compatibility criteria). The Luders’s compatibility criteria in higher finite dimensional spaces is rather difficult to check for any prospective fixed quantum state. Some part of our results cover the famous contractive fixed point results of Banach, Kannan and Chatterjea.

The most prominent Business Process Model Abstraction (BPMA) use case is the construction of the process “quick view” for rapidly comprehending a complex process. Some researchers propose process abstraction methods to aggregate the activities on the basis of their semantic similarity. One important clustering technique used in these methods is traditional k-means cluster analysis which so far is an unsupervised process without any priori information, and most of the techniques aggregate the activities only according to business semantics without considering the requirement of an order-preserving model transformation. The paper proposes a BPMA method based on semi-supervised clustering which chooses the initial clusters based on the refined process structure tree and designs constraints by combining the control flow consistency of the process and the semantic similarity of the activities to guide the clustering process. To be more precise, the constraint function is discovered by mining from a process model collection enriched with subprocess relations. The proposed method is validated by applying it to a process model repository in use. In an experimental validation, the proposed method is compared to the traditional k-means clustering (parameterized with randomly chosen initial clusters and an only semantics-based distance measure), showing that the approach closely approximates the decisions of the involved modelers to cluster activities. As such, the paper contributes to the development of modeling support for effective process model abstraction, facilitating the use of business process models in practice. [ABSTRACT FROM AUTHOR]

It is well known that the transformation semigroup on a nonempty set X, which is denoted by T(X), is regular, but its subsemigroups do not need to be. Consider a finite ordered set X = (X; ≤) whose order forms a path with alternating orientation. For a nonempty subset Y of X, two subsemigroups of T(X) are studied. Namely, the semigroup OT(X, Y) = {α ∈ T(X) | α is order-preserving and Xα ⊆ Y} and the semigroup OS(X, Y) = {α ∈ T(X) | α is order-preserving and Y α ⊆ Y}. In this paper, we characterize ordered sets having a coregular semigroup OT(X, Y) and a coregular semigroup OS(X, Y), respectively. Some characterizations of regular semigroups OT(X, Y) and OS(X, Y) are given. We also describe coregular and regular elements of both OT(X, Y) and OS(X, Y). [ABSTRACT FROM AUTHOR]

Copyright of Journal of Guangdong University of Technology is the property of Journal of Guangdong University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let Xn = {1, 2,..., n) with its natural order and let Tn be the full transformation semigroup on Xn. A map α ∈ Tn is said to be order-preserving if, for all x, y ∈ Xn, x ≤ y x⇒ xα ≤ yα. The map α ∈ Tn is said to be a contraction if, for all x, y ∈ Xn, |xα - yα| ≤ |x - y|. Let CTn and OCTn denote, respectively, subsemigroups of all contraction maps and all order-preserving contraction maps in Tn. In this paper we present characterisations of Green's relations on CTn and starred Green's relations on both CTn and OCTn. [ABSTRACT FROM AUTHOR]

Let be the semigroup of all singular order-preserving mappings on the finite set . It is known that is generated by its set of idempotents of rank , and its rank and idempotent rank are and , respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank in . We also calculate its rank and idempotent rank. [ABSTRACT FROM AUTHOR]

In this paper we give presentations for the monoid $${\mathcal {DP}}_n$$ of all partial isometries on $$\{1,\ldots ,n\}$$ and for its submonoid $${\mathcal {ODP}}_n$$ of all order-preserving partial isometries. [ABSTRACT FROM AUTHOR]

In this paper, we determine the relative rank of the semigroup T(X; Y) of all transformations on a finite set X with restricted range Y modulo the semigroup of all extensions of the bijections on Y, modulo the idempotent order-preserving transformations in T(X; Y), and modulo the semigroup of all order-preserving trans- formations in T(X; Y). [ABSTRACT FROM AUTHOR]

It is well-known that the transformation semigroup is regular, but their subsemigroups need not be. A fence is an ordered set that the order forms a path with alternating orientation. Consider as the base set of a fence . Two subsemigroups of are studied. Namely, the semigroup of all order-decreasing self-mappings of and the semigroup of all order-preserving self-mappings of . In this paper, we obtain that is a coregular subsemigroup of . A characterization of regular subsemigroups of is given, that is, is regular if and only if . Finally, we discuss the regularity of elements in . [ABSTRACT FROM AUTHOR]

We describe the dynamics of a system of two reaction-diffusion equations with 1-homogeneous non-linearity. We show that either an order-preserving property holds and can be used in order to determine the limiting behaviour in some (invariant) sets or the long time behaviour of all solutions can be described by looking at one scalar reaction-diffusion equation only. Submitted November 15, 2000. Published September 27, 2002. Math Subject Classifications: 35K57 Key Words: Reaction-diffusion equations; order-preserving; reduction to one equation; oscillation number

Graph layouts and visualizations have been at the forefront of graph drawing research for decades, consequently leading to aesthetic heuristics that not only generate better visualizations and aesthetically appealing graphs but also improve readability and understanding of the graphs. A variety of approaches examines aesthetics of nodes, edges, or graph layout, and related readability metrics. Improving the known bounds on two aesthetic requirements (area and aspect ratio) for planar straight-line order-preserving grid drawings of binary trees is presented in an algorithm that uses a separation approach. The new bounds are optimal in area and aspect ratio, where the optimum values are linear and 1:1 respectively. [ABSTRACT FROM AUTHOR]

In this paper we consider various classes of monoids of transformations of a finite chain, including those of transformations that preserve or reverse either the order or the orientation. In line with Howie and McFadden [24], we complete the study of the ranks (and of idempotent ranks, when applicable) of all their ideals. [ABSTRACT FROM AUTHOR]

For n ∈ ℕ, let On be the semigroup of all singular order-preserving mappings on [n]= {1,2,...,n}. For each nonempty subset A of [n], let On(A) = {α ∈ On: (∀ k ∈A) kα ≤ k} be the semigroup of all order-preserving and A-decreasing mappings on [n]. In this paper it is shown that On(A) is an abundant semigroup with n-1풟*-classes. Moreover, On(A) is idempotent-generated and its idempotent rank is 2n-2- |A\ {n}|. Further, it is shown that the rank of On(A) is equal to n-1 if 1 ∈ A, and it is equal to n otherwise. [ABSTRACT FROM AUTHOR]

Let X be a finite or infinite chain and let ${\mathcal{O}}(X)$ be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of ${\mathcal{O}}(X)$ and Green's relations on ${\mathcal{O}}(X)$. In fact, more generally, if Y is a nonempty subset of X and ${\mathcal{O}}(X,Y)$ is the subsemigroup of ${\mathcal{O}}(X)$ of all elements with range contained in Y, we characterize the largest regular subsemigroup of ${\mathcal{O}}(X,Y)$ and Green's relations on ${\mathcal{O}}(X,Y)$. Moreover, for finite chains, we determine when two semigroups of the type ${\mathcal {O}}(X,Y)$ are isomorphic and calculate their ranks. [ABSTRACT FROM AUTHOR]

In the present article, motivated by the work of Zhu et al. [10], we introduce a new class of expansive mapping termed as (Ψ, f)s-expansive and (f; g; h)s-weakly expansive mappings. These two newly proposed expansive mappings are utilized to establish some common fixed point theorems in the set up of ordered partial b-metric spaces. Examples are provided to verify the effectiveness and applicability of our main results. [ABSTRACT FROM AUTHOR]

In this paper we describe locally maximal idempotent-generated subsemigroups of finite orientation-preserving singular partial transformation semigroups SPOPn and obtain a complete classification. We also obtain a classification of locally maximal idempotent-generated subsemigroups of finite order-preserving singular partial transformation semigroups with respect to ≤k. [ABSTRACT FROM AUTHOR]

Abstract: In order to compare different genome sequences, an alignment-free method has been proposed. Considering the essential property of sequence is sequentiality, we define a compound transformation which transforms a genome sequence into a sparse 16 by matrix M based on 16 kinds of 2-mer (dinucleotides). Furthermore, we found the transformation above-mentioned is an order-preserving transformation (OPT). Based on the theory of matrix analysis, we derive a 16-dimensional vector to characterize a genome sequence via singular value decomposition (SVD) of M . Finally, we analyze the similarities among multiple sequences from 20 eutherian species. The experiment results show that our approach performs well in the field of sequence analysis. [Copyright &y& Elsevier]

In this paper we describe the maximal idempotent-generated subsemigroups of the finite orientation-preserving singular partial transformation semigroup SPOP and obtain their complete classification. We also obtain a classification of the maximal idempotent-generated subsemigroups of the finite order-preserving singular partial transformation semigroup $\mathit{PO}^{k}_{n}$ with respect to ≤. [ABSTRACT FROM AUTHOR]

Let O be the order-preserving transformation semigroup on X. For an arbitrary integer r such that 1≤ r≤ n−2, we completely describe the maximal regular subsemibands of the semigroup K( n, r)={ α∈ O:|im( α)|≤ r}. We also formulate the cardinal number of such subsemigroups. [ABSTRACT FROM AUTHOR]

Traditional database query processing relies on three types of algorithms for join and for grouping operations. For joins, index nested loops join exploits an index on its inner input, merge join exploits sorted inputs, and hash join exploits differences in the sizes of the join inputs. For grouping, an index-based algorithm has been used in the past whereas today sort- and hash-based algorithms prevail. Cost-based query optimization chooses the most appropriate algorithm for each query and for each operation. Unfortunately, mistaken algorithm choices during compile-time query optimization are common yet expensive to investigate and to resolve. Our goal is to end mistaken choices among join algorithms and among grouping algorithms by replacing the three traditional types of algorithms with a single one. Like merge join, this new join algorithm exploits sorted inputs. Like hash join, it exploits different input sizes for unsorted inputs. In fact, for unsorted inputs, the cost functions for recursive hash join and for hybrid hash join have guided our search for the new join algorithm. In consequence, the new join algorithm can replace both merge join and hash join in a database management system. The in-memory components of the new join algorithm employ indexes. If the database contains indexes for one (or both) of the inputs, the new join can exploit persistent indexes instead of temporary in-memory indexes. Using database indexes to find matching input records, the new join algorithm can also replace index nested loops join. In addition to join operations, a very similar algorithm supports grouping ('group by' queries in SQL) and duplicate elimination. For unsorted inputs, candidate output records take on the role of one of the inputs in a join operation. Our goal is to define a single grouping algorithm that can replace grouping by repeated index searches, by sorting, and by hashing. In other words, our goal is to end mistaken algorithm choices not only for joins and other binary matching operations but also for grouping and other unary matching operations in database query processing. Finally, these new algorithms can be instrumental for efficient and robust data processing in a map-reduce environment, because 'map' and 'reduce' operations are similar in essentials to join and grouping operations. Results from an implementation of the core algorithm are reported. [ABSTRACT FROM AUTHOR]

In this note we consider various classes of monoids of transformations on a finite chain, in particular of transformations that preserve or reverse either the order or the orientation. Being finite monoids we are naturally interested in computing both their cardinals and their idempotent numbers. Fibonacci and Lucas numbers play an essential role in the last computations. [ABSTRACT FROM AUTHOR]

In this paper we present a division theorem for the pseudovariety of semigroups 햮햣 (햮햱) generated by all semigroups of order-preserving or order-reversing (orientation-preserving or orientation-reversing) full transformations on a finite chain. [ABSTRACT FROM AUTHOR]

Abstract: Let be a lattice, let be a meet-closed subset of and let be a function. We characterize the matrix divisibility of the join matrix by the meet matrix in the ring in terms of the usual divisibility in , and we present two algorithms for constructing certain classes of meet-closed sets such that divides . As an example we present the lattice-theoretic structure of all meet-closed sets with at most five elements possessing the matrix divisibility property. Finally, we show that our methods solve some open problems in the divisor lattice, concerning the divisibility of GCD and LCM matrices. [Copyright &y& Elsevier]