In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. Semigroups may be considered a special case of magmaswhere the operation is associative, or as a generalization of groupswithout requiring the existence of an identity element or inverses. If the semigroup operation is commutative, then the semigroup is called a commutative semigroup or less often than in the analogous case of groups it may be called an abelian semigroup.
A monoid is an algebraic structure intermediate between groups and semigroups, and is a semigroup having an identity elementthus obeying all but one of the axioms of a group; existence of inverses is not required of a monoid. A natural example is strings with concatenation as the binary operation, and the empty string as the identity element.
Restricting to non-empty strings gives an example of a semigroup that is not a monoid. Positive integers with addition form a commutative semigroup that is not a monoid, whereas the non-negative integers do form a monoid. A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory.
Semigroups should not be confused with quasigroupswhich are a generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups preserve from groups a notion of division.
Division in semigroups or in monoids is not possible in general. The formal study of semigroups began in the early 20th century. Early results include a Cayley theorem for semigroups realizing any semigroup as transformation semigroupin which arbitrary functions replace the role of bijections from group theory.
Congruences on *-Simple Type A I-Semigroups
Some other techniques for studying semigroups, like Green's relationsdo not resemble anything in group theory. The theory of finite semigroups has been of particular importance in theoretical computer science since the s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In probability theorysemigroups are associated with Markov processes.[Discrete Mathematics] Modular Arithmetic
In partial differential equationsa semigroup is associated to any equation whose spatial evolution is independent of time. There are numerous special classes of semigroupssemigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention: regular semigroupsorthodox semigroupssemigroups with involutioninverse semigroups and cancellative semigroups.
There also interesting classes of semigroups that do not contain any groups except the trivial group ; examples of the latter kind are bands and their commutative subclass— semilatticeswhich are also ordered algebraic structures. More succinctly, a semigroup is an associative magma. Left and right identities are both called one-sided identities.
A semigroup may have one or more left identities but no right identity, and vice versa. A two-sided identity or just identity is an element that is both a left and right identity. Semigroups with a two-sided identity are called monoids. A semigroup may have at most one two-sided identity. If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup.
If a semigroup has both a left identity and a right identity, then it has a two-sided identity which is therefore the unique one-sided identity. Similarly, every magma has at most one absorbing elementwhich in semigroup theory is called a zero. This notion is defined identically as it is for groups. In terms of this operation, a subset A is called. If A is both a left ideal and a right ideal then it is called an ideal or a two-sided ideal.
If S is a semigroup, then the intersection of any collection of subsemigroups of S is also a subsemigroup of S. So the subsemigroups of S form a complete lattice.
An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a commutative semigroup, when it exists, is a group. Green's relationsa set of five equivalence relations that characterise the elements in terms of the principal ideals they generate, are important tools for analysing the ideals of a semigroup and related notions of structure.The aim of this paper is to investigate the lattices of group fuzzy congruences and normal fuzzy subsemigroups on E -inversive semigroups.
We prove that group fuzzy congruences and normal fuzzy subsemigroups determined each other in E -inversive semigroups. The investigation of fuzzy sets is initiated by Zadeh in [ 1 ]. As special fuzzy sets, fuzzy congruences on groups and semigroups have been extensively studied by many authors. InKuroki [ 2 ] introduced fuzzy congruences on a group and characterized fuzzy congruences on a group in terms of fuzzy normal subgroups.
InSamhan [ 3 ] studied the modularity condition in the fuzzy congruence lattice of a semigroup and derived that the fuzzy congruence lattice of a group is modular. In the same year, Al-Thukair [ 4 ] described the fuzzy congruences of an inverse semigroup and obtained a one-one correspondence between fuzzy congruence pairs and fuzzy congruences on an inverse semigroup.
Moreover, Kuroki also studied the fuzzy congruences on inverse semigroups in [ 5 ] in which the notion of group congruences of a semigroup is provided. Das [ 6 ] considered the lattice of fuzzy congruences in an inverse semigroup by kernel-trace approaches.
InAjmal and Thomas considered the lattice structures of fuzzy congruences on a group and the lattice structures of fuzzy subgroups and fuzzy normal subgroups in a group in [ 7 ] and proved that the lattice of fuzzy normal subgroups of a group is modular in [ 8 ]. InKim and Bae [ 9 ] studied the fuzzy congruences of groups and obtained several results which are analogs of some basic theorems of group theory. Also, Xie [ 10 ] studied the so-called fuzzy Rees congruences on semigroups in Several authors investigated fuzzy congruences for some special classes of semigroups.
InZhang [ 11 ] characterized the group fuzzy congruences on a regular semigroup by some fuzzy subsemigroups. InTan [ 12 ] investigated fuzzy congruences of regular semigroups and proved that the lattice of fuzzy congruences on a regular semigroup is a disjoint union of some modular sublattices of the lattice. Recently, Li and Liu [ 13 ] characterized fuzzy good congruences of left semiperfect abundant semigroups and obtained sufficient and necessary conditions for an abundant semigroup to be left semiperfect.
The class of E -inversive semigroups is a very wide class of semigroups which contains groups, inverse semigroups, and regular semigroups as proper subclasses and some kinds of crisp congruences on this class of semigroups have been investigated extensively; see [ 1415 ] for example.
Inspired by the above facts, it is natural to study the fuzzy congruences on E -inversive semigroups. In fact, [ 16 ] has done some works in this aspect. In this paper, we shall investigate the lattices of group fuzzy congruences and normal fuzzy subsemigroups on an E -inversive semigroup.
The notions of group t -fuzzy congruences and normal fuzzy subsemigroups with tip t on E -inversive semigroups are proposed and some properties of them are given.A semigroup generalizes a monoid in that a semigroup need not have an identity element. It also originally generalized a group a monoid with all inverses in that no element had to have an inversethus the name semigroup.
By definition, a semigroup is an associative magma. A semigroup with an identity element is called a monoid. A group is then a monoid in which every element has an inverse element. Semigroups must not be confused with quasigroupswhich are sets with a not necessarily associative binary operation such that division is always possible. The formal study of semigroups began in the early 20th century. Semigroups are important in many areas of mathematics because they are the abstract algebraic underpinning of "memoryless" systems: time-dependent systems that start from scratch at each iteration.
In applied mathematicssemigroups are fundamental models for linear time-invariant systems. In partial differential equationsa semigroup is associated to any equation whose spatial evolution is independent of time. The theory of finite semigroups has been of particular importance in theoretical computer science since the s because of the natural link between finite semigroups and finite automata.
In probability theorysemigroups are associated with Markov processes Template:Harv. Template:Algebraic structures.
More succinctly, a semigroup is an associative magma. If it has both a left identity and a right identity, a semigroup and indeed magma has at most one identity elementwhich is then two-sided. A semigroup with identity is called a monoid. A semigroup may have multiple left identities but no right identity or vice versa.
Similarly, every magma has at most one absorbing elementwhich in semigroup theory is called a zero. Analogous to the above construction, for every semigroup Sone can define S 0a semigroup with 0 that embeds S. In terms of this operations, a subset A is called. If A is both a left ideal and a right ideal then it is called an ideal or a two-sided ideal. If S is a semigroup, then the intersection of any collection of subsemigroups of S is also a subsemigroup of S.Thanks for helping us catch any problems with articles on DeepDyve.
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Let S be an eventually regular semigroup. The extensively P-partial congruence pairs and P-partial congruence pairs for S are defined. Semigroup Forum — Springer Journals. Enjoy affordable access to over 18 million articles from more than 15, peer-reviewed journals. Get unlimited, online access to over 18 million full-text articles from more than 15, scientific journals.
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Submitting a report will send us an email through our customer support system. Submit report Close. Recommended Articles Loading References Concepts of congruence, morphic image and substructure for biordered sets.Let S be a regular semigroup and E S be the set of its idempotents. We characterize regular semigroups S in which all S e; f or all S a are right zero semigroups respectively are trivial in several ways including weak versions of compatibility of the natural order.
This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Auinger : Free objects in joins of strict inverse and completely simple semigroups.
London Math. Auinger : The congruence lattice of a strict regular semigroup. Pure Appl. Algebra 81— Blyth and M. Gomes : On the compatibility of the natural order on a regular semigroup. Royal Soc. Edinburgh A9479— Howie : An introduction to semigroup theory. Academic Press, London, Google Scholar.
On sandwich sets and congruences on regular semigroups
Nambooripad : Structure of regular semigroups. Nambooripad : The natural partial order on a regular semigroup. Edinburgh Math. Petrich : Introduction to semigroups. Merrill, Columbus, Petrich : Inverse semigroups. Wiley, New York, Trotter : Congruence extensions in regular semigroups. Algebra— Venkatesan : Right left inverse semigroups.
Algebra 31— Download references. Reprints and Permissions. Petrich, M. On sandwich sets and congruences on regular semigroups. Czech Math J 56, 27—46 Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries.
The convention followed in this article will be that of writing a function on the right of its argument, e. Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner  in the Soviet Union in and by Gordon Preston in the United Kingdom in Partial transformations had already been studied in the context of pseudogroups.
With the addition of this empty transformation, the composition of partial transformations of a set becomes an everywhere-defined associative binary operation. There are a number of equivalent characterisations of an inverse semigroup S : . There is therefore a simple characterisation of Green's relations in an inverse semigroup: . Unless stated otherwise, E S will denote the semilattice of idempotents of an inverse semigroup S.
The natural partial order is compatible with both multiplication and inversion, that is, . In a groupthis partial order simply reduces to equality, since the identity is the only idempotent. In a symmetric inverse semigroup, the partial order reduces to restriction of mappings, i.
On E Sthe natural partial order becomes:. If E S is finite and forms a chain i. The homomorphic image of an inverse semigroup is an inverse semigroup; the inverse of an element is always mapped to the inverse of the image of that element.
One of the earliest results proved about inverse semigroups was the Wagner—Preston Theoremwhich is an analogue of Cayley's Theorem for groups :. Wagner—Preston Theorem. Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup, and with image closed under the inverse operation on partial bijections. Conversely, any subsemigroup of the symmetric inverse semigroup closed under the inverse operation is an inverse semigroup. Hence a semigroup S is isomorphic to a subsemigroup of the symmetric inverse semigroup closed under inverses if and only if S is an inverse semigroup.
In the set of all group congruences on a semigroup Sthe minimal element for the partial order defined by inclusion of sets need not be the smallest element. One class of inverse semigroups that has been studied extensively over the years is the class of E -unitary inverse semigroups: an inverse semigroup S with semilattice E of idempotents is E - unitary if, for all e in E and all s in S.
Then the following are equivalent: . McAlister's Covering Theorem. Every inverse semigroup S has a E-unitary cover; that is there exists an idempotent separating surjective homomorphism from some E-unitary semigroup T onto S. Central to the study of E -unitary inverse semigroups is the following construction. A McAlister triple is used to define the following:. McAlister's P-Theorem. Conversely, every E -unitary inverse semigroup is isomorphic to one of this type. An inverse semigroup is said to be F -inverse if every element has a unique maximal element above it in the natural partial order, i.
Every F -inverse semigroup is an E -unitary monoid. McAlister's covering theorem has been refined by M. Lawson to:. Every inverse semigroup has an F -inverse cover.An eventually regular semigroup is a semigroup in which some power of any element is regular.
The minimum group congruence on an eventually regular semigroup is investigated by means of weak inverse. Furthermore, some properties of the minimum group congruence on an eventually regular semigroup are characterized. Recall that a semigroup is said to be eventually regular if each of its elements which has some power is regular. From the definition we conclude that eventually regular semigroups generalize both regular and finite semigroups.
Edwards [ 2 ] was successful in showing that many results for regular semigroups can be obtained for eventually regular semigroups. The strategy to study eventually regular semigroups is to generalize known results for regular semigroups to eventually regular semigroups.
Group congruences on regular semigroups have been investigated by many algebraists. Latorre [ 3 ] explored group congruences on regular semigroups extensively and gave the representation of group congruences on regular semigroups. Hanumantha [ 4 ] generalized the results in [ 3 ] for regular semigroups to eventually regular semigroups. In this paper, the author explores the minimum group congruences on eventually regular semigroups by means of weak inverses. A new representation of the minimum group congruence on an eventually regular semigroup is given.
Furthermore, group congruences on eventually regular semigroups are described in the same technique. Lemma 2. Remark 2. Theorem 3. Then the following statements are true. Lemma 3. Proof of Theorem 3. It follows from Lemma 3. It follows from Lemma 2. We, by Lemma 3. We now turn to proving that the converse holds. The proof is then completed. Corollary 3. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We will be providing unlimited waivers of publication charges for accepted articles related to COVID Sign up here as a reviewer to help fast-track new submissions. Academic Editor: G. Received 29 May Accepted 03 Jul Published 14 Aug