In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a semigroup

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

 is a nonempty set together with an associative binary operation

Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

. A special class of semigroups is a class

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...

 of semigroup

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

s satisfying additional properties

Property (philosophy)

In modern philosophy, logic, and mathematics a property is an attribute of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property however differs from individual objects in that...

 or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity

Commutativity

In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

 property that ab = ba for all elements a and b in the semigroup.
The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large variety of special classes of semigroups have been defined though not all of them have been studied equally intensively.

In the algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

ic theory

Theory

The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...

 of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s of the underlying set. The underlying sets are not assumed to carry any other mathematical structure

Structure

Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...

s like order or topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

.

As in any algebraic theory, one of the main problems of the theory of semigroups is the classification of all semigroups and a complete description of their structure. In the case of semigroups, since the binary operation is required to satisfy only the associativity property the problem of classification is considered extremely difficult. Descriptions of structures have been obtained for certain special classes of semigroups. For example the structure of the sets of idempotents of regular semigroups is completely known. Structure descriptions are presented in terms of better known types of semigroups. The best known type of semigroup is the group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

.

A (necessarily incomplete) list of various special classes of semigroups is presented below. To the extent possible the defining properties are formulated in terms of the binary operations in the semigroups. The references point to the locations from where the defining properties are sourced.

Notations

In describing the defining properties of the various special classes of semigroups, the following notational conventions are adopted.

Notations

Notation	Meaning
S	Arbitrary semigroup
E	Set of idempotents in S
G	Group of units in S
X	Arbitrary set
a, b, c	Arbitrary elements of S
x, y, z	Specific elements of S
e, f. g	Arbitrary elements of E
h	Specific element of E
l, m, n	Arbitrary positive integers
j, k	Specific positive integers
0	Zero element of S
1	Identity element of S
S1	S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S
L, R, H, D, J	Green's relations
La, Ra, Ha, Da, Ja	Green classes containing a
a ≤L b a ≤R b a ≤H b	S1a ⊆ S1b aS1 ⊆ bS1 S1a ⊆ S1b and aS1 ⊆ bS1

List of special classes of semigroups

List of special classes of semigroups

Terminology	Defining property	Reference(s)
Finite semigroup	S is a finite set.
Empty semigroup Empty semigroup In mathematics, a semigroup with no elements is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation. However not all authors insist...	S =
Trivial semigroup Trivial semigroup In mathematics, a trivial semigroup is a semigroup for which the cardinality of the underlying set is one. The number of distinct nonisomorphic semigroups with one element is one...	Cardinality of S is 1.
Monoid Monoid In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...	1 ∈ S	Gril p.3
Band Band (algebra) In mathematics, a band is a semigroup in which every element is idempotent . Bands were first studied and named by ; the lattice of varieties of bands was described independently in the early 1970s by Biryukov, Fennemore and Gerhard... (Idempotent semigroup)	a2 = a	C&P  p.4
Idempotent semigroup (Band Band (algebra) In mathematics, a band is a semigroup in which every element is idempotent . Bands were first studied and named by ; the lattice of varieties of bands was described independently in the early 1970s by Biryukov, Fennemore and Gerhard... )	a2 = a	C&P  p.4
Semilattice Semilattice In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...	a2 = a ab = ba	C&P p.24
Commutative semigroup	ab = ba	C&P p.3
Archimedean Archimedean property In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...  commutative semigroup	ab = ba There exists x and k such that a = xbk.	C&P p.131
Nowhere commutative semigroup Nowhere commutative semigroup In mathematics, a nowhere commutative semigroup is a semigroup S such that, for all a and b in S, if ab = ba then a = b...	ab = ba   ⇒   a = b	C&P p.26
Left weakly commutative	There exist x and k such that (ab)k = bx.	Nagy p.59
Right weakly commutative	There exist x and k such that (ab)k = xb.	Nagy p.59
Weakly commutative	There exist x and j such that (ab)j = bx. There exist y and k such that (ab)k = yb.	Nagy p.59
Conditionally commutative semigroup	If ab = ba then axb = bxa for all x.	Nagy p.77
R-commutative semigroup	ab R ba	Nagy p.69–71
RC-commutative semigroup	R-commutative and conditionally commutative	Nagy p.93–107
L-commutative semigroup	ab L ba	Nagy p.69–71
LC-commutative semigroup	L-commutative and conditionally commutative	Nagy p.93–107
H-commutative semigroup	ab H ba	Nagy p.69–71
Quasi-commutative semigroup	ab = (ba)k for some k.	Nagy p.109
Right commutative semigroup	xab = xba	Nagy p.137
Left commutative semigroup	abx = bax	Nagy p.137
Externally commutative semigroup	axb = bxa	Nagy p.175
Medial semigroup	xaby = xbay	Nagy p.119
E-k semigroup (k fixed)	(ab)k = akbk	Nagy p.183
Exponential Exponential Exponential may refer to any of several mathematical topics related to exponentiation, including:*Exponential function, also:**Matrix exponential, the matrix analogue to the above*Exponential decay, decrease at a rate proportional to value...  semigroup	(ab)m = ambm for all m	Nagy p.183
WE-k semigroup (k fixed)	There is a positive integer j depending on the couple (a,b) such that (ab)k+j = akbk (ab)j = (ab)jakbk	Nagy p.199
Weakly exponential Exponential Exponential may refer to any of several mathematical topics related to exponentiation, including:*Exponential function, also:**Matrix exponential, the matrix analogue to the above*Exponential decay, decrease at a rate proportional to value...  semigroup	WE-m for all m	Nagy p.215
Cancellative semigroup Cancellative semigroup In mathematics, a cancellative semigroup is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality of the form a · b = a · c, where · is a binary operation, one can cancel the element a and deduce the equality b = c...	ax = ay   ⇒   x = y xa = ya   ⇒   x = y	C&P p.3
Right cancellative semigroup Cancellative semigroup In mathematics, a cancellative semigroup is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality of the form a · b = a · c, where · is a binary operation, one can cancel the element a and deduce the equality b = c...	xa = ya   ⇒   x = y	C&P p.3
Left cancellative semigroup Cancellative semigroup In mathematics, a cancellative semigroup is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality of the form a · b = a · c, where · is a binary operation, one can cancel the element a and deduce the equality b = c...	ax = ay   ⇒   x = y	C&P p.3
E-inversive Inversive Inversive activities are processes which self internalise the action concerned. For example a person who has an Inversive personality internalises his emotions from any exterior source...  semigroup	There exists x such that ax ∈ E.	C&P p.98
Regular semigroup Regular semigroup A regular semigroup is a semigroup S in which every element is regular, i.e., for each element a, there exists an element x such that axa = a. Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations.- Origins...	There exists x such that axa =a.	C&P p.26
Intra-regular semigroup	There exist x and y such that xa2y = a.	C&P p.121
Left regular semigroup	There exists x such that xa2 = a.	C&P p.121
Right regular semigroup	There exists x such that a2x = a.	C&P p.121
Completely regular semigroup Completely regular semigroup In mathematics, a completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. The class of completely regular semigroups forms an important subclass of the class of regular semigroups, the class of inverse semigroups being another such subclass... (Clifford semigroup)	Ha is a group.	Gril p.75
k-regular semigroup (k fixed)	There exists x such that akxak = ak.	Hari
π-regular semigroup (Quasi regular semigroup, Eventually regular semigroup)	There exists k and x (depending on a) such that akxak = ak.	Edwa
Eventually regular semigroup (π-regular semigroup, Quasi regular semigroup)	There exists k and x (depending on a) such that akxak = ak.	Edwa
Quasi-regular semigroup (π-regular semigroup, Eventually regular semigroup)	There exists k and x (depending on a) such that akxak = ak.	Shum
Primitive semigroup	If 0 ≠ e and f = ef = fe then e = f.	C&P p.26
Unit regular semigroup	There exists u in G such that aua = a.	Tvm
Strongly unit regular semigroup	There exists u in G such that aua = a. e D f ⇒ f = v-1ev for some v in G.	Tvm
Orthodox semigroup	There exists x such that axa = a. E is a subsemigroup of S.	Gril p.57
Inverse semigroup Inverse semigroup In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...	There exists unique x such that axa = a and xax = x.	C&P p.28
Left inverse semigroup (R-unipotent)	Ra contains a unique h.	Gril p.382
Right inverse semigroup (L-unipotent)	La contains a unique h.	Gril p.382
Locally inverse semigroup (Pseudoinverse semigroup)	There exists x such that axa = a. E is a pseudosemilattice.	Gril p.352
M-inversive semigroup	There exist x and y such that baxc = bc and byac = bc.	C&P p.98
Pseudoinverse semigroup (Locally inverse semigroup)	There exists x such that axa = a. E is a pseudosemilattice.	Gril p.352
Abundant semigroups	The classes L*a and R*a, where a L* b if ac = ad ⇔ bc = bd and a R* b if ca = da ⇔ cb = db, contain idempotents.	Chen
Rpp-semigroup (Right principal projective semigroup)	The class L*a, where a L* b if ac = ad ⇔ bc = bd, contains at least one idempotent.	Shum
Lpp-semigroup (Left principal projective semigroup)	The class R*a, where a R* b if ca = da ⇔ cb = db, contains at least one idempotent.	Shum
Null semigroup Null semigroup In mathematics, a null semigroup is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero...   (Zero semigroup)	0 ∈ S ab = 0	C&P p.4
Zero semigroup (Null semigroup Null semigroup In mathematics, a null semigroup is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero... )	0 ∈ S ab = 0	C&P p.4
Left zero semigroup	ab = a	C&P p.4
Right zero semigroup	ab = b	C&P p.4
Unipotent semigroup	E is singleton.	C&P p.21
Left reductive semigroup	If xa = xb for all x implies a = b.	C&P p.9
Right reductive semigroup	If ax = bx for all x implies a = b.	C&P p.4
Reductive semigroup	If xa = xb for all x implies a = b. If ax = bx for all x implies a = b.	C&P p.4
Separative semigroup	ab = a2 = b2   ⇒   a = b	C&P p.130–131
Reversible semigroup	Sa ∩ Sb ≠ Ø aS ∩ bS ≠ Ø	C&P p.34
Right reversible semigroup	Sa ∩ Sb ≠ Ø	C&P p.34
Left reversible semigroup	aS ∩ bS ≠ Ø	C&P p.34
Aperiodic semigroup (Groupfree semigroup, Combinatorial semigroup)	ab = ba ak is in a subgroup of S for some k. Every nonempty subset of E has an infimum. Every subgroup of S is trivial.	Gril p.119
ω-semigroup	E is countable descending chain under the order a ≤H b	Gril p.233–238
Clifford semigroup (Completely regular semigroup Completely regular semigroup In mathematics, a completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. The class of completely regular semigroups forms an important subclass of the class of regular semigroups, the class of inverse semigroups being another such subclass... )	Ha is a group.	Gril p.211–215
Left Clifford semigroup (LC-semigroup)	aS ⊆ Sa	Shum
Right Clifford semigroup (RC-semigroup)	Sa ⊆ aS	Shum
LC-semigroup (Left Clifford semigroup)	aS ⊆ Sa	Shum
RC-semigroup (Right Clifford semigroup)	Sa ⊆ aS	Shum
Orthogroup	Ha is a group. E is a subsemigroup of S	Shum
Combinatorial semigroup (Aperiodic semigroup, Groupfree semigroup)	ab = ba ak is in a subgroup of S for some k. Every nonempty subset of E has an infimum. Every subgroup of S is trivial.	Gril p.119
Complete commutative semigroup	ab = ba ak is in a subgroup of S for some k. Every nonempty subset of E has an infimum.	Gril p.110
Nilsemigroup	0 ∈ S ak = 0 for some k.	Gril p.99
Elementary semigroup	ab = ba S = G ∪ N where G is a group, N is a nilsemigroup or a one-element semigroup. N is ideal of S. Identity of G is 1 of S and zero of N is 0 of S.	Gril p.111
E-unitary semigroup	There exists unique x such that axa = a and xax = x. ea = e   ⇒   a ∈ E	Gril p.245
Finitely presented semigroup	S has a presentation ( X; R ) in which X and R are finite.	Gril p.134
Fundamental semigroup	Equality on S is the only congruence contained in H.	Gril p.88
Groupfree semigroup (Aperiodic semigroup, Combinatorial semigroup)	ab = ba ak is in a subgroup of S for some k. Every nonempty subset of E has an infimum. Every subgroup of S is trivial.	Gril p.119
Idempotent generated semigroup	S is equal to the semigroup generated by E.	Gril p.328
Locally finite semigroup	Every finitely generated subsemigroup of S is finite.	Gril p.161
N-semigroup	ab = ba There exists x and a positive integer n such that a = xbn. ax = ay   ⇒   x = y xa = ya   ⇒   x = y E = Ø	Gril p.100
L-unipotent semigroup (Right inverse semigroup)	La contains a unique e.	Gril p.362
R-unipotent semigroup (Left inverse semigroup)	Ra contains a unique e.	Gril p.362
Left simple semigroup	La = S	Gril p.57
Right simple semigroup	Ra = S	Gril p.57
Subelementary semigroup	ab = ba S = C ∪ N where C is a cancellative semigroup, N is a nilsemigroup or a one-element semigroup. N is ideal of S. Zero of N is 0 of S. For x, y in S and c in C, cx = cy implies that x = y.	Gril p.134
Symmetric semigroup (Full transformation semigroup Transformation semigroup In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup is associated with the composite of the two corresponding... )	Set of all mappings of X into itself with composition of mappings as binary operation.	C&P p.2
Weakly reductive semigroup	If xz = yz and zx = zy for all z in S then x = y.	C&P p.11
Right unambiguous semigroup	If x, y ≥R z then x ≥R y or y ≥R x.	Gril p.170
Left unambiguous semigroup	If x, y ≥L z then x ≥L y or y ≥L x.	Gril p.170
Unambiguous semigroup	If x, y ≥R z then x ≥R y or y ≥R x. If x, y ≥L z then x ≥L y or y ≥L x.	Gril p.170
Left 0-unambiguous	0∈ S 0 ≠ x ≤L y, z   ⇒   y ≤L z or z ≤L y	Gril p.178
Right 0-unambiguous	0∈ S 0 ≠ x ≤R y, z   ⇒   y ≤L z or z ≤R y	Gril p.178
0-unambiguous semigroup	0∈ S 0 ≠ x ≤L y, z   ⇒   y ≤L z or z ≤L y 0 ≠ x ≤R y, z   ⇒   y ≤L z or z ≤R y	Gril p.178
Left Putcha semigroup	a ∈ bS1   ⇒   an ∈ b2S1 for some n.	Nagy p.35
Right Putcha semigroup	a ∈ S1b   ⇒   an ∈ S1b2 for some n.	Nagy  p.35
Putcha semigroup	a ∈ S1b S1   ⇒   an ∈ S1a2S1 for some positive integer n	Nagy  p.35
Bisimple semigroup (D-simple semigroup)	Da = S	C&P p.49
0-bisimple semigroup	0 ∈ S S - {0} is a D-class of S.	C&P p.76
Completely simple semigroup	There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A. There exists h in E such that whenever hf = f and fh = f we have h = f.	C&P p.76
Completely 0-simple semigroup	0 ∈ S S2 ≠ 0 If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0. There exists h in E such that whenever hf = f and fh = f we have h = f or h = 0.	C&P p.76
D-simple semigroup (Bisimple semigroup)	Da = S	C&P p.49
Semisimple semigroup	Let J(a) = S1aS1, I(a) = J(a) – Ja. Each Rees factor semigroup J(a)/I(a) is 0-simple or simple.	C&P p.71–75
Simple semigroup	There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A.	C&P p.5
0-simple semigroup	0 ∈ S S2 ≠ 0 If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0.	C&P p.67
Left 0-simple semigroup	0 ∈ S S2 ≠ 0 If A ⊆ S is such that SA ⊆ A then A = 0.	C&P p.67
Right 0-simple semigroup	0 ∈ S S2 ≠ 0 If A ⊆ S is such that AS ⊆ A then A = 0.	C&P p.67
Cyclic semigroup  (Monogenic semigroup Monogenic semigroup In mathematics, a monogenic semigroup is a semigroup generated by a set containing only a single element. Monogenic semigroups are also called cyclic semigroups.-Structure:... )	S = { w, w2, w3, ... } for some w in S	C&P p.19
Monogenic semigroup Monogenic semigroup In mathematics, a monogenic semigroup is a semigroup generated by a set containing only a single element. Monogenic semigroups are also called cyclic semigroups.-Structure:... (Cyclic semigroup)	S = { w, w2, w3, ... } for some w in S	C&P p.19
Periodic semigroup	{ a, a2, a3, ... } is a finite set.	C&P p.20
Bicyclic semigroup Bicyclic semigroup In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup. The first published description of this object was given by Evgenii Lyapin in 1953. Alfred H...	1 ∈ S S generated by { x1, x2 } with x1x2 = 1.	C&P p.43–46
Full transformation semigroup TX (Symmetric semigroup)	Set of all mappings Map (mathematics) In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...  of X into itself with composition of mappings as binary operation Binary operation In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division.... .	C&P p.2
Rectangular semigroup	Whenever three of ax, ay, bx, by are equal, all four are equal.	C&P p.97
Symmetric inverse semigroup IX	The semigroup of one-to-one partial transformations Partial function In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...  of X.	C&P p.29
Brandt semigroup Brandt semigroup In mathematics, Brandt semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideal and which are also inverse semigroups. They are build in the same way that completely 0-simple semigroups:...	0 ∈ S ( ac = bc ≠ 0 or ca = cb ≠ 0 )   ⇒   a = b ( ab ≠ 0 and bc ≠ 0 )   ⇒   abc ≠ 0 If a ≠ 0 there exist unique x, y, z, such that xa = a, ay = a, za = y. ( e ≠ 0 and f ≠ 0 )   ⇒   eSf ≠ 0.	C&P p.101
Free semigroup Free semigroup In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences of zero or more elements from A. It is usually denoted A∗. The identity element is the unique sequence of zero elements, often called the empty string and denoted by ε or λ, and the...  FX	Set of finite sequences of elements of X with the operation ( x1, ... , xm ) ( y1, ... , yn ) = ( x1, ... , xm, y1, ... , yn )	Gril p.18
Rees matrix Matrix (mathematics) In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...  semigroup	G0 a group G with 0 adjoined. P : Λ × I → G0 a map. Define an operation in I × G0 × Λ by ( i, g, λ ) ( j, h, μ ) = ( i, g P( λ, μ ) h, μ ). ( I, G0, Λ )/( I × { 0 } × Λ ) is the Ress matrix semigroup M0 ( G0; I, Λ ; P ).	C&P p.88
Semigroup of linear transformation Linear transformation In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0... s	Semigroup of linear transformation Linear transformation In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0... s of a vector space Vector space A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...  V over a field Field (mathematics) In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...  F under composition of functions.	C&P p.57
Semigroup of binary relation Binary relation In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of... s BX	Set of all binary relation Binary relation In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of... s on X under composition Composition of relations In mathematics, the composition of binary relations is a concept of forming a new relation from two given relations R and S, having as its most well-known special case the composition of functions.- Definition :...	C&P p.13
Numerical semigroup Numerical semigroup In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of addition of integers. Also, the integer 0 must be an element of the semigroup...	0 ∈ S ⊆ N = { 0,1,2, ... } under + . N - S is finite	Delg
Semigroup with involution Semigroup with involution In mathematics, in semigroup theory, an involution in a semigroup is a transformation of the semigroup which is its own inverse and which is an anti-automorphism of the semigroup. A semigroup in which an involution is defined is called a semigroup with involution... (*-semigroup)	There exists an unary operation a → a* in S such that a** = a and (ab)* = b*a*.	Howi
*-semigroup (Semigroup with involution Semigroup with involution In mathematics, in semigroup theory, an involution in a semigroup is a transformation of the semigroup which is its own inverse and which is an anti-automorphism of the semigroup. A semigroup in which an involution is defined is called a semigroup with involution... )	There exists an unary operation a → a* in S such that a** = a and (ab)* = b*a*.	Howi
Baer–Levi semigroup	Semigroup of one-to-one transformations f of X such that X − f ( X ) is infinite.	C&P II Ch.8
U-semigroup	There exists a unary operation a → a’ in S such that ( a’)’ = a.	Howi p.102
I-semigroup	There exists a unary operation a → a’ in S such that ( a’)’ = a and aa’a = a.	Howi p.102
Group Group (mathematics) In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...	There exists h such that ah = ha = a. There exists x (depending on a) such that ax = xa = h.