DOCSLIB.ORG

A Divertimento on Monadplus and Nondeterminism1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Divertimento on Monadplus and Nondeterminism1 A Divertimento on MonadPlus and Nondeterminism1 Tarmo Uustalu Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia Abstract In the Haskell community, there is a controversy about what the laws of the MonadPlus type constructor class ought to be. We suggest that there is no single universal correct answer, however there is a universal method. Important classes of notions of finitary nondeterminism are captured by what we call monads of semigroups and monads of monoids, but also by monads of different specializations of semigroups and monoids. Some of these specializations of monads are exotic and amusing too. Keywords: monads, algebraic operations, semigroups, nondeterminism 1. Introduction Monads are a widely used in functional programming, especially Haskell, as an ab- straction of computational effects. The mathematics of monads is very well understood, as they are a central concept of category theory. But Haskell's standard library features also a more specific type constructor class MonadPlus to cover a particular variety of effects, namely different notions of finitary nondeterminism. Oddly, the mathematics of MonadPlus has been unclear. The question of what the \right" laws of MonadPlus ought to be has been open from the beginning and there is no commonly agreed answer. Moreover, until the recent work by Rivas et al. [12], there have been no attempts to justify any proposal by mathematical considerations. In this note, we pursue one principled way of addressing this question. We take inspiration from the algebraic operations approach of Plotkin and Power [11]. We sug- gest that there is no single universal correct answer, rather there is a universal method. Important classes of notions of finitary nondeterministic choice are captured by what we call monads of semigroups and monads of monoids, but also by monads of different specializations (varieties) of semigroups and monoids. We argue that such special types of monads, uniformly defined following a certain pattern, are well-motivated, since they enjoy neat mathematical properties. At the same time, one to be specific about what idea of finitary nondeterminism is had in mind, i.e., what exactly is meant to be ob- servable about nondeterministic computations, following a clear method. Some of these 1Dedicated to Jos´eNuno Oliveira on the occasion of his 60th birthday. Email address: [email protected] (Tarmo Uustalu) Preprint submitted to Elsevier September 20, 2015 specializations of monads are quite exotic, but all of them are instructive and fun to study. I have titled this piece a divertimento, since I consider it lighthearted; certainly it was intended so. I have tried to make it entertaining, yet serious, and thus fit for a festive social function like a birthday event is. And above all I hope I found a beautiful theme which is easy to memorize. I have learned a lot from you, Jos´eNuno, and the thing I value above all from our interactions is your overflowing enthusiasm about beautiful mathematics and music. The paper is organized as follows. First, in Section 2, we review the MonadPlus class of Haskell. In Section 3, we introduce monads of semigroups and show that this is a well-motivated concept with neat mathematical properties that also corresponds to a natural class of notions of finitary nondeterminism. In Section 4, we introduce monads of monoids as a similar concept with analogous properties. In Section 5, we discuss monads of some specializations of semigroups to capture narrower classes of notions of nondeterminism (where less and less operational detail is observable). In Section 6, we point to the closest related work, to conclude in Section 7. We assume that the reader is knowledgeable about monads and their application to programming with effects. To keep the presentation simple, especially concerning examples, we use Set as the base category throughout the presentation. But any category with finite products would work for most of our purposes. 2. The MonadPlus class of Haskell The Control.Monad module of Haskell's standard library provides a type class called MonadPlus [3]. Its definition says that a MonadPlus instance is a Monad instance equipped with two polymorphic functions mzero and mplus. class Monad m => MonadPlus m where mzero :: m a mplus :: m a -> m a -> m a MonadPlus is thus a subclass of Monad. It is intended as an abstraction of notions of finitary nondeterminism rather than of notions of just any type of effectful computation. mzero is the operation of nullary choice (choice with no options) and mplus is binary choice. But MonadPlus has also been applied to organize search with backtracking, sometimes regarded as \don't know" nondeterminism, in opposition to normal (demonic) \don't care" nondeterminism. The most important intended MonadPlus instance is the list monad, with the empty list and concatenation of lists as mzero and mplus. Other wanted instances include, e.g., the combination of the list monad with the state monad for some state set. The maybe monad is seen by some as an intended instance, as a monad to use for backtracking, while others want to exclude it, perhaps arguing that it should belong to a different type class with member functions of the same types but subject to a different set of laws. A remarkable controversy about MonadPlus is that there is no agreement about the \right" laws (this question has been discussed in countless mailinglist threads, wiki pages etc., see, e.g., [13, 5, 4]). It seems generally accepted that mzero and mplus should obey the laws of a monoid, i.e., the left and right unital laws and associativity: 2 mzero `mplus` c == c c `mplus` mzero == c (c `mplus` c') `mplus` c'' == c `mplus` (c' `mplus` c'') And also, mzero should be a left zero of bind and bind should right distribute over mplus2: mzero >>= k == mzero (c `mplus` c') >>= k == (c >>= k) `mplus` (c' >>= k) (Right distributivity fails for the maybe monad.) Some require in addition that mzero be a right zero and bind left distribute over mplus: c >>= \ _ -> mzero == mzero c >>= \ a -> (k a `mplus` k' a) == (c >>= k) `mplus` (c >>= k') But left distributivity fails already for the list monad. Among further laws that one may want to impose for some purposes are commuta- tivity and idempotence of mplus. c `mplus` c' == c' `mplus` c c `mplus` c == c Of course these fail for the list monad. But lists may be used as representations for finite sets and under this view commutativity and idempotence hold on the level of denotations. Yet another law sometimes suggested is the following so-called left catch law. return a `mplus` c == return a This is satisfied by the maybe monad but falsified by the list monad. Together with right distributivity, left catch gives k a `mplus` (c >>= k) == k a which is unreasonably strong. In this paper, we aim to give a well-motivated method for answering the question of what the laws of the MonadPlus class should be. We concentrate first on the case of where only mplus is present and justify a relatively weak axiomatization (only associativity together with as little more as one can feasibly get away with|it turns out that there are very good reasons to include right distributivity). Then we add mzero. And then we look at a number of laws that can be included optionally in order to capture narrower classes of notions of nondeterminism. 2Confusingly, this law is called left distributivity in most documents about MonadPlus. 3 3. Monads of semigroups We follow the algebraic operations approach of Plotkin and Power [11]. We take interest in monads that \support" some algebraic operations. We say that a monad of semigroups is a monad T = (T; η; µ) with a family of maps ⊕X : TX ×TX ! TX natural in X such that ⊕ is associative and µ is right distributive over ⊕: ⊕ ×TX µX ×µX (TX × TX) × TX X / TX × TX T (TX) × T (TX) / TX × TX αT X;T X;T X TX × (TX × TX) ⊕X ⊕TX ⊕X TX×⊕X ⊕X µX TX × TX / TX T (TX) / TX (c ⊕ c0) ⊕ c00 = c ⊕ (c0 ⊕ c00) µ (c ⊕ c0) = µ c ⊕ µ c0 The right distributivity law can also be expressed in terms of (−)∗ (the Kleisli extension operation) rather than µ and then ensures also naturality of ⊕: k∗ (c ⊕ c0) = k∗ c ⊕ k∗ c0 We claim that monads of semigroups are canonical in several ways and make a good abstraction of those notions of finitary nondeterminism based on binary choice where the temporal order of binary choices in a computation (the interior of the computation tree) is not observable. It is immediate from the definition that a monad of semigroups (T; ⊕) delivers semi- groups in the sense that, for any X,(T X; ⊕X ) is a semigroup|this is ensured by the associativity law. Moreover, right distributivity says that µX is a semigroup homomor- phism between (T (TX); µTX ) and (T X; µX ). Abstractly, while a monad is nothing but a monoid in the monoidal category ([Set; Set]; Id; ·), a monad of semigroups is the same thing as a right near-semiring in the right near-semiring category ([Set; Set]; ×; Id; ·). This becomes clear when we write the laws of ⊕ on the level of functors: ⊕×T µ×µ (T × T ) × T / T × T T · T × T · T / T × T αT;T;T T × (T × T ) ⊕X (T × T ) · T ⊕ T ×⊕ ⊕·T ⊕ µ T × T / TX T · T / T But the more interesting properties of a monad of semigroups pertain to its Eilenberg- Moore category and Kleisli category. Also, the initial monad of semigroups stands out by additional special properties. 4 The Eilenberg-Moore category of a monad of semigroups First, algebras of a monad of semigroups (T; ⊕) are semigroups in the sense of there being a functor F : EM(T) ! Sgrp preserving carriers: F EM(T) / Sgrp FF z FF zz FF z U FF zz U F# }zz C The semigroup structure : A × A ! A for a T-algebra (A; α) is defined by ηA×ηA ⊕A α = A × A / TA × TA / TA / A The converse is also true.
Load more

Recommended publications
  • Interaction Laws of Monads and Comonads
    Interaction Laws of Monads and Comonads Shin-ya Katsumata1 , Exequiel Rivas2 , and Tarmo Uustalu3;4 1 National Institute of Informatics, Tokyo, Japan 2 Inria Paris, France 3 Dept. of Computer Science, Reykjavik University, Iceland 4 Dept. of Software Science, Tallinn University of Technology, Estonia [email protected], [email protected], [email protected] Abstract. We introduce and study functor-functor and monad-comonad interaction laws as math- ematical objects to describe interaction of effectful computations with behaviors of effect-performing machines. Monad-comonad interaction laws are monoid objects of the monoidal category of functor- functor interaction laws. We show that, for suitable generalizations of the concepts of dual and Sweedler dual, the greatest functor resp. monad interacting with a given functor or comonad is its dual while the greatest comonad interacting with a given monad is its Sweedler dual. We relate monad-comonad interaction laws to stateful runners. We show that functor-functor interaction laws are Chu spaces over the category of endofunctors taken with the Day convolution monoidal struc- ture. Hasegawa's glueing endows the category of these Chu spaces with a monoidal structure whose monoid objects are monad-comonad interaction laws. 1 Introduction What does it mean to run an effectful program, abstracted into a computation? In this paper, we take the view that an effectful computation does not perform its effects; those are to be provided externally. The computation can only proceed if placed in an environment that can provide its effects, e.g, respond to the computation's requests for input, listen to its output, resolve its nondeterministic choices by tossing a coin, consistently respond to its fetch and store commands.
    [Show full text]
  • Varieties Generated by Unstable Involution Semigroups with Continuum Many Subvarieties
    Comptes Rendus Mathématique 356 (2018) 44–51 Varieties generated by unstable involution semigroups with continuum many subvarieties Edmond W.H. Lee Department of Mathematics, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale, FL 33314, USA a b s t r a c t Over the years, several finite semigroups have been found to generate varieties with continuum many subvarieties. However, finite involution semigroups that generate varieties with continuum many subvarieties seem much rarer; in fact, only one example—an inverse semigroup of order 165—has so far been published. Nevertheless, it is shown in the present article that there are many smaller examples among involution semigroups that are unstable in the sense that the varieties they generate contain some involution semilattice with nontrivial unary operation. The most prominent examples are the unstable finite involution semigroups that are inherently non-finitely based, the smallest ones of which are of order six. It follows that the join of two finitely generated varieties of involution semigroups with finitely many subvarieties can contain continuum many subvarieties. 45 1. Introduction By the celebrated theorem of Oates and Powell[23], the variety VAR Ggenerated by any finite groupGcontains finitely many subvarieties. In contrast, the variety VAR S generated by a finite semigroup S can contain continuum many sub- varieties. Aprominent source of such semigroups, due to Jackson[9], is the class of inherently non-finitely based finite semigroups. The multiplicative matrix semigroup B1 = 00 , 10 , 01 , 00 , 00 , 10 , 2 00 00 00 10 01 01 commonly called the Brandt monoid, is the most well-known inherently non-finitely based semigroup[24].
    [Show full text]
  • Free Split Bands Francis Pastijn Marquette University, [email protected]
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by epublications@Marquette Marquette University e-Publications@Marquette Mathematics, Statistics and Computer Science Mathematics, Statistics and Computer Science, Faculty Research and Publications Department of 6-1-2015 Free Split Bands Francis Pastijn Marquette University, [email protected] Justin Albert Marquette University, [email protected] Accepted version. Semigroup Forum, Vol. 90, No. 3 (June 2015): 753-762. DOI. © 2015 Springer International Publishing AG. Part of Springer Nature. Used with permission. Shareable Link. Provided by the Springer Nature SharedIt content-sharing initiative. NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page. Free Split Bands Francis Pastijn Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI Justin Albert Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI Abstract: We solve the word problem for the free objects in the variety consisting of bands with a semilattice transversal. It follows that every free band can be embedded into a band with a semilattice transversal. Keywords: Free band, Split band, Semilattice transversal 1 Introduction We refer to3 and6 for a general background and as references to terminology used in this paper. Recall that a band is a semigroup where every element is an idempotent. The Green relation is the least semilattice congruence on a band, and so every band is a semilattice of its -classes; the -classes themselves form rectangular bands.5 We shall be interested in bands S for which the least semilattice congruence splits, that is, there exists a subsemilattice of which intersects each -class in exactly one element.
    [Show full text]
  • Bands with Isomorphic Endomorphism Semigroups* BORISM
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA %, 548-565 (1985) Bands with Isomorphic Endomorphism Semigroups* BORISM. SCHEIN Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701 Communicated by G. B. Preston Received March 10. 1984 Let End S denote the endomorphism semigroup of a semigroup S. If S and Tare bands (i.e., idempotent semigroups), then every isomorphism of End S onto End T is induced by a uniquely determined bijection of S onto T. This bijection either preserves both the natural order relation and O-quasi-order relation on S or rever- ses both these relations. If S is a normal band, then the bijection induces isomorphisms of all rectangular components of S onto those of T, or it induces anti-isomorphisms of the rectangular components. If both 5’ and T are normal bands, then S and T are isomorphic, or anti-isomorphic, or Sg Y x C and Tz Y* x C, where Y is a chain considered as a lower semilattice, Y* is a dual chain, and C is a rectangular band, or S= Y x C, Tr Y* x C*rr, where Y, Y*, and C are as above and Cow is antiisomorphic to C. 0 1985 Academic Press, Inc. The following well-known problem is considered in this paper. Suppose S and T are two mathematical structures of the same type. Let End S denote the semigroup of all self-morphisms of S into itself. Suppose that End T is an analogous semigroup for T and the semigroups End S and End T are isomorphic.
    [Show full text]
  • Arxiv:1508.05446V2 [Math.CO] 27 Sep 2018 02,5B5 16E10
    CELL COMPLEXES, POSET TOPOLOGY AND THE REPRESENTATION THEORY OF ALGEBRAS ARISING IN ALGEBRAIC COMBINATORICS AND DISCRETE GEOMETRY STUART MARGOLIS, FRANCO SALIOLA, AND BENJAMIN STEINBERG Abstract. In recent years it has been noted that a number of combi- natorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent pa- per, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple mod- ules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of ex- amples is a left regular band structure on the face poset of a CAT(0) cube complex.
    [Show full text]
  • Semilattice Sums of Algebras and Mal'tsev Products of Varieties
    Mathematics Publications Mathematics 5-20-2020 Semilattice sums of algebras and Mal’tsev products of varieties Clifford Bergman Iowa State University, [email protected] T. Penza Warsaw University of Technology A. B. Romanowska Warsaw University of Technology Follow this and additional works at: https://lib.dr.iastate.edu/math_pubs Part of the Algebra Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ math_pubs/215. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Mathematics at Iowa State University Digital Repository. It has been accepted for inclusion in Mathematics Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Semilattice sums of algebras and Mal’tsev products of varieties Abstract The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider when this quasivariety is a variety. The main result shows that if V is a strongly irregular variety with no nullary operations, and S is a variety, of the same type as V, equivalent to the variety of semilattices, then the Mal’tsev product V ◦ S is a variety. It consists precisely of semilattice sums of algebras in V. We derive an equational basis for the product from an equational basis for V. However, if V is a regular variety, then the Mal’tsev product may not be a variety. We discuss examples of various applications of the main result, and examine some detailed representations of algebras in V ◦ S.
    [Show full text]
  • Free Split Bands Francis Pastijn Marquette University, [email protected]
    Marquette University e-Publications@Marquette Mathematics, Statistics and Computer Science Mathematics, Statistics and Computer Science, Faculty Research and Publications Department of 6-1-2015 Free Split Bands Francis Pastijn Marquette University, [email protected] Justin Albert Marquette University, [email protected] Accepted version. Semigroup Forum, Vol. 90, No. 3 (June 2015): 753-762. DOI. © 2015 Springer International Publishing AG. Part of Springer Nature. Used with permission. Shareable Link. Provided by the Springer Nature SharedIt content-sharing initiative. NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page. Free Split Bands Francis Pastijn Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI Justin Albert Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI Abstract: We solve the word problem for the free objects in the variety consisting of bands with a semilattice transversal. It follows that every free band can be embedded into a band with a semilattice transversal. Keywords: Free band, Split band, Semilattice transversal 1 Introduction We refer to3 and6 for a general background and as references to terminology used in this paper. Recall that a band is a semigroup where every element is an idempotent. The Green relation is the least semilattice congruence on a band, and so every band is a semilattice of its -classes; the -classes themselves form rectangular bands.5 We shall be interested in bands S for which the least semilattice congruence splits, that is, there exists a subsemilattice of which intersects each -class in exactly one element.
    [Show full text]
  • REGULAR SEMIGROUPS WHICH ARE SUBDIRECT PRODUCTS of a BAND and a SEMILATTICE of GROUPS by MARIO PETRICH (Received 27 April, 1971)
    REGULAR SEMIGROUPS WHICH ARE SUBDIRECT PRODUCTS OF A BAND AND A SEMILATTICE OF GROUPS by MARIO PETRICH (Received 27 April, 1971) 1. Introduction and summary. In the study of the structure of regular semigroups, it is customary to impose several conditions restricting the behaviour of ideals, idempotents or elements. In a few instances, one may represent them as subdirect products of some much more restricted types of regular semigroups, e.g., completely (0-) simple semigroups, bands, semilattices, etc. In particular, studying the structure of completely regular semigroups, one quickly distinguishes certain special cases of interest when these semigroups are represented as semilattices of completely simple semigroups. In fact, this semilattice of semigroups may be built in a particular way, idempotents may form a subsemigroup, Jf may be a congruence, and soon. Instead of making some arbitrary choice of these conditions, we consider, in a certain sense, the converse problem, by starting with regular semigroups which are subdirect products of a band and a semilattice of groups. This choice, in turn, may seem arbitrary, but it is a natural one in view of the following abstract characterization of such semigroups: they are bands of groups and their idempotents form a subsemigroup. It is then quite natural to consider various special cases such as regular semigroups which are subdirect products of: a band and a group, a rectangular band and a semilattice of groups, etc. For each of these special cases, we establish (a) an abstract characterization in terms of completely regular semigroups satisfying some additional restrictions, (b) a construction from the component semigroups in the subdirect product, (c) an isomorphism theorem in terms of the represen- tation in (b), (d) a relationship among the congruences on an arbitrary regular semigroup which yield a quotient semigroup of the type under study.
    [Show full text]
  • The Structure and Characterizaions of Normal Clifford Semirings Jiao Han, Gang Li
    IAENG International Journal of Applied Mathematics, 51:3, IJAM_51_3_05 ______________________________________________________________________________________ The Structure and Characterizaions of Normal Clifford Semirings Jiao Han, Gang Li • Abstract—In this paper, we define normal Clifford semirings, (Rα; +); α 2 D, since E (S) is a normal band, the (S; +) which are generalizations of rectangular Clifford semirings. is a normal orthogroup in which each maximal subgroup is We also give the necessary and sufficient conditions for a + semiring to be normal Clifford semiring and the spined product abelian. So S= D is the distributive lattice D. It is clear that decomposition of normal Clifford semirings. We also discuss a + special case of this kind of semirings, that is strong distributive s D t () V +(s) + s = t + V +(t) lattices of rectangular rings. () (V +(s) + s) \ (t + V +(t)) 6= ;: Index Terms—rectangular rings, normal Clifford semirings, + Distributive lattice congruence, normal band. Due to D is the distributive lattice congruence on semiring + + + S, we get s D s2, st D ts, s(s + t) D s. Foy any c 2 sV +(s), I. INTRODUCTION there exists x 2 V +(s) such that c = sx, from the law of A semiring S = (S; +; •) is an algebra with two binary distribution, we have operation ”+”, ”•” such that the additive reduct (S:+) and s2 + sx + s2 = s(s + x + s) = ss = s2; the multiplicative reduct (S; •) are semigroups connected sx + s2 + sx = s(x + s + x) = sx; by ring-like distributive laws. A semiring S = (S; +; •) is called idempotent semiring, if (8s 2 S)s + s = s = s • s. then sV +(s) ⊆ V +(s2).
    [Show full text]
  • Semilattice of Bisimple Regular Semigroups
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Number 2, May 1971 SEMILATTICE OF BISIMPLE REGULAR SEMIGROUPS H. R. KRISHNA IYENGAR Abstract. The main purpose of this paper is to show that a regular semigroup S is a semilattice of bisimple semigroups if and only if it is a band of bisimple semigroups and that this holds if and only if 3) is a congruence on S. It is also shown that a quasiregular semigroup 5 which is a rectangular band of bisimple semigroups is itself bisimple. In [3, Theorem 4.4] it was shown that a semigroup S is a semi- lattice of simple semigroups if and only if it is a union of simple semi- groups. The purpose of this paper is to obtain corresponding results for a semigroup which is a semilattice of bisimple regular semi- groups. Unfortunately, a semilattice of bisimple semigroups need not be a union of bisimple semigroups as illustrated by a simple w-semi- group constructed by Munn [S]. However, we get some equivalent conditions for such semigroups. In particular we show that a regular semigroup is a semilattice of bisimple semigroups if and only if it is a band of bisimple semigroups. 1. Equivalent conditions. In this section we consider a set of equiv- alent conditions for a semigroup 5 to be a semilattice of bisimple semigroups. We adopt the terminology and notation of [2]. Lemma 1.1. Let S be a semilattice ß of semigroups Sa and let D be a Si-class of S. Then, either Sai^D = □ or DQSa.
    [Show full text]
  • Partial Order in the Normal Category Arising from Normal Bands
    Partial Order in the Normal Category Arising from Normal Bands C S Preenu University College Thiruvananthapuram Research Centre University of Kerala A R Rajan Former Professor of Mathematics University of Kerala K S Zeenath Former Professor of Mathematics School of Distance Education University of Kerala Abstract The notion of normal category was introduced by KSS Namboori- pad in connection with the study of the structure of regular semigroups using cross connections[4]. It is an abstraction of the category of prin- cipal left ideals of a regular semigroup. A normal band is a semigroup B satisfying a2 = a and abca = acba for all a,b,c ∈ B. Since the normal bands are regular semigroups, the category L(B) of principal left ideals of a normal band B is a normal category. One of the special properties of this category is that the morphism sets admit a partial order compatible with the composition of morphisms. In this article we derive several properties of this partial order and obtain a new characterization of this partial order. Keywords: Partial order, Normal categories, Normal bands. AMS Subject Classification : 20M17 arXiv:2105.01135v1 [math.CT] 3 May 2021 1 Preliminaries In this section we discuss some basic concepts of normal categories, normal bands and the normal category derived from normal bands. We follow [2, 4] for other notations and definitions which are not mentioned. 1.1 Normal Categories We begin with the category theory, some basic definitions and notations. In a category C, the class of objects is denoted by vC and C itself is used 1 to denote the morphism class.
    [Show full text]
  • Varieties of P-Restriction Semigroups
    Varieties of P -restriction semigroups Peter R. Jones October 27, 2011 Abstract The restriction semigroups, in both their one-sided and two-sided versions, have arisen in various fashions, meriting study for their own sake. From one historical perspective, as `weakly E-ample' semigroups, the definition revolves around a `designated set' of commuting idempotents, better thought of as projections. This class includes the inverse semigroups in a natural fashion. In a recent paper, the author introduced P -restriction semigroups in order to broaden the notion of `projection' (thereby encompassing the regular ∗-semigroups). That study is continued here from the varietal perspective introduced for restriction semigroups by V. Gould. The relationship between varieties of regular ∗-semigroups and varieties of P - restriction semigroups is studied. In particular, a tight relationship exists between varieties of orthodox ∗-semigroups and varieties of `orthodox' P -restriction semigroups, leading to concrete descriptions of the free orthodox P -restriction semigroups and related structures. Specializing further, new, elementary paths are found for descriptions of the free restriction semigroups, in both the two-sided and one-sided cases. In [13], the author introduced P -restriction semigroups as a common generalization of the restriction semigroups (or `weakly E-ample' semigroups) and the regular ∗-semigroups, defining them as bi-unary semigroups { semigroups with two additional unary operations, + and ∗ { satisfying a set of simple identities. The projections in such a semigroup are the elements of the form x+ (or, equivalently, x∗). If (S; ·;−1 ) is a regular ∗-semigroup and + and ∗ are defined respectively by x+ = xx−1 and x∗ = x−1x, then its bi-unary reduct is a P -restriction semigroup.
    [Show full text]