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In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity 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 collection of special classes of semigroups have been defined though not all of them have been studied equally intensively.
In the algebraic theory 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 subsets of the underlying set. The underlying sets are not assumed to carry any other mathematical structures like order or topology.
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.
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.
In describing the defining properties of the various special classes of semigroups, the following notational conventions are adopted.
Notation | Meaning |
---|---|
S | Arbitrary semigroup |
E | Set of idempotents in S |
G | Group of units in S |
I | Minimal ideal of S |
V | Regular elements of 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 |
v, w | Arbitrary elements of V |
0 | Zero element of S |
1 | Identity element of S |
S1 | S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S |
a ≤Lb a ≤Rb a ≤Hb a ≤Jb | S1a ⊆ S1b aS1 ⊆ bS1 S1a ⊆ S1b and aS1 ⊆ bS1 S1aS1 ⊆ S1bS1 |
L, R, H, D, J | Green's relations |
La, Ra, Ha, Da, Ja | Green classes containing a |
The only power of x which is idempotent. This element exists, assuming the semigroup is (locally) finite. See variety of finite semigroups for more information about this notation. | |
The cardinality of X, assuming X is finite. | |
For example, the definition xab = xba should be read as:
The third column states whether this set of semigroups forms a variety. And whether the set of finite semigroups of this special class forms a variety of finite semigroups. Note that if this set is a variety, its set of finite elements is automatically a variety of finite semigroups.
Terminology | Defining property | Variety of finite semigroup | Reference(s) |
---|---|---|---|
Finite semigroup |
|
| |
Empty semigroup |
| No | |
Trivial semigroup |
|
| |
Monoid |
| No | Gril p. 3 |
Band (Idempotent semigroup) |
|
| C&P p. 4 |
Rectangular band |
|
| Fennemore |
Semilattice | A commutative band, that is:
|
| |
Commutative semigroup |
|
| C&P p. 3 |
Archimedean commutative semigroup |
| C&P p. 131 | |
Nowhere commutative semigroup |
| C&P p. 26 | |
Left weakly commutative |
| Nagy p. 59 | |
Right weakly commutative |
| Nagy p. 59 | |
Weakly commutative | Left and right weakly commutative. That is:
| Nagy p. 59 | |
Conditionally commutative semigroup |
| Nagy p. 77 | |
R-commutative semigroup |
| Nagy p. 69–71 | |
RC-commutative semigroup |
| Nagy p. 93–107 | |
L-commutative semigroup |
| Nagy p. 69–71 | |
LC-commutative semigroup |
| Nagy p. 93–107 | |
H-commutative semigroup |
| Nagy p. 69–71 | |
Quasi-commutative semigroup |
| Nagy p. 109 | |
Right commutative semigroup |
| Nagy p. 137 | |
Left commutative semigroup |
| Nagy p. 137 | |
Externally commutative semigroup |
| Nagy p. 175 | |
Medial semigroup |
| Nagy p. 119 | |
E-k semigroup (k fixed) |
|
| Nagy p. 183 |
Exponential semigroup |
|
| Nagy p. 183 |
WE-k semigroup (k fixed) |
| Nagy p. 199 | |
Weakly exponential semigroup |
| Nagy p. 215 | |
Right cancellative semigroup |
| C&P p. 3 | |
Left cancellative semigroup |
| C&P p. 3 | |
Cancellative semigroup | Left and right cancellative semigroup, that is
| C&P p. 3 | |
''E''-inversive semigroup (E-dense semigroup) |
| C&P p. 98 | |
Regular semigroup |
| C&P p. 26 | |
Regular band |
|
| Fennemore |
Intra-regular semigroup |
| C&P p. 121 | |
Left regular semigroup |
| C&P p. 121 | |
Left-regular band |
|
| Fennemore |
Right regular semigroup |
| C&P p. 121 | |
Right-regular band |
|
| Fennemore |
Completely regular semigroup |
| Gril p. 75 | |
(inverse) Clifford semigroup |
|
| Petrich p. 65 |
k-regular semigroup (k fixed) |
| Hari | |
Eventually regular semigroup (π-regular semigroup, Quasi regular semigroup) |
| Edwa Shum Higg p. 49 | |
Quasi-periodic semigroup, epigroup, group-bound semigroup, completely (or strongly) π-regular semigroup, and many other; see Kela for a list) |
| Kela Gril p. 110 Higg p. 4 | |
Primitive semigroup |
| C&P p. 26 | |
Unit regular semigroup |
| Tvm | |
Strongly unit regular semigroup |
| Tvm | |
Orthodox semigroup |
| Gril p. 57 Howi p. 226 | |
Inverse semigroup |
| C&P p. 28 | |
Left inverse semigroup (R-unipotent) |
| Gril p. 382 | |
Right inverse semigroup (L-unipotent) |
| Gril p. 382 | |
Locally inverse semigroup (Pseudoinverse semigroup) |
| Gril p. 352 | |
M-inversive semigroup |
| C&P p. 98 | |
Abundant semigroup |
| Chen | |
Rpp-semigroup (Right principal projective semigroup) |
| Shum | |
Lpp-semigroup (Left principal projective semigroup) |
| Shum | |
Null semigroup (Zero semigroup) |
|
| C&P p. 4 |
Left zero semigroup |
|
| C&P p. 4 |
Left zero band | A left zero semigroup which is a band. That is:
|
| |
Left group |
| C&P p. 37, 38 | |
Right zero semigroup |
|
| C&P p. 4 |
Right zero band | A right zero semigroup which is a band. That is:
|
| Fennemore |
Right group |
| C&P p. 37, 38 | |
Right abelian group |
| Nagy p. 87 | |
Unipotent semigroup |
|
| C&P p. 21 |
Left reductive semigroup |
| C&P p. 9 | |
Right reductive semigroup |
| C&P p. 4 | |
Reductive semigroup |
| C&P p. 4 | |
Separative semigroup |
| C&P p. 130–131 | |
Reversible semigroup |
| C&P p. 34 | |
Right reversible semigroup |
| C&P p. 34 | |
Left reversible semigroup |
| C&P p. 34 | |
Aperiodic semigroup |
| ||
ω-semigroup |
| Gril p. 233–238 | |
Left Clifford semigroup (LC-semigroup) |
| Shum | |
Right Clifford semigroup (RC-semigroup) |
| Shum | |
Orthogroup |
| Shum | |
Complete commutative semigroup |
| Gril p. 110 | |
Nilsemigroup (Nilpotent semigroup) |
|
| |
Elementary semigroup |
| Gril p. 111 | |
E-unitary semigroup |
| Gril p. 245 | |
Finitely presented semigroup |
| Gril p. 134 | |
Fundamental semigroup |
| Gril p. 88 | |
Idempotent generated semigroup |
| Gril p. 328 | |
Locally finite semigroup |
|
| Gril p. 161 |
N-semigroup |
| Gril p. 100 | |
L-unipotent semigroup (Right inverse semigroup) |
| Gril p. 362 | |
R-unipotent semigroup (Left inverse semigroup) |
| Gril p. 362 | |
Left simple semigroup |
| Gril p. 57 | |
Right simple semigroup |
| Gril p. 57 | |
Subelementary semigroup |
| Gril p. 134 | |
Symmetric semigroup (Full transformation semigroup) |
| C&P p. 2 | |
Weakly reductive semigroup |
| C&P p. 11 | |
Right unambiguous semigroup |
| Gril p. 170 | |
Left unambiguous semigroup |
| Gril p. 170 | |
Unambiguous semigroup |
| Gril p. 170 | |
Left 0-unambiguous |
| Gril p. 178 | |
Right 0-unambiguous |
| Gril p. 178 | |
0-unambiguous semigroup |
| Gril p. 178 | |
Left Putcha semigroup |
| Nagy p. 35 | |
Right Putcha semigroup |
| Nagy p. 35 | |
Putcha semigroup |
| Nagy p. 35 | |
Bisimple semigroup (D-simple semigroup) |
| C&P p. 49 | |
0-bisimple semigroup |
| C&P p. 76 | |
Completely simple semigroup |
| C&P p. 76 | |
Completely 0-simple semigroup |
| C&P p. 76 | |
D-simple semigroup (Bisimple semigroup) |
| C&P p. 49 | |
Semisimple semigroup |
| C&P p. 71–75 | |
: Simple semigroup |
|
| |
0-simple semigroup |
| C&P p. 67 | |
Left 0-simple semigroup |
| C&P p. 67 | |
Right 0-simple semigroup |
| C&P p. 67 | |
Cyclic semigroup (Monogenic semigroup) |
|
| C&P p. 19 |
Periodic semigroup |
|
| C&P p. 20 |
Bicyclic semigroup |
| C&P p. 43–46 | |
Full transformation semigroup TX (Symmetric semigroup) |
| C&P p. 2 | |
Rectangular band |
|
| Fennemore |
Rectangular semigroup |
| C&P p. 97 | |
Symmetric inverse semigroup IX |
| C&P p. 29 | |
Brandt semigroup |
| C&P p. 101 | |
Free semigroup FX |
| Gril p. 18 | |
Rees matrix semigroup |
| C&P p.88 | |
Semigroup of linear transformations |
| C&P p.57 | |
Semigroup of binary relations BX |
| C&P p.13 | |
Numerical semigroup |
| Delg | |
Semigroup with involution (*-semigroup) |
| Howi | |
Baer–Levi semigroup |
| C&P II Ch.8 | |
U-semigroup |
| Howi p.102 | |
I-semigroup |
| Howi p.102 | |
Semiband |
| Howi p.230 | |
Group |
|
| |
Topological semigroup |
|
| Pin p. 130 |
Syntactic semigroup |
| Pin p. 14 | |
: the R-trivial monoids |
|
| Pin p. 158 |
: the L-trivial monoids |
|
| Pin p. 158 |
: the J-trivial monoids |
|
| Pin p. 158 |
: idempotent and R-trivial monoids |
|
| Pin p. 158 |
: idempotent and L-trivial monoids |
|
| Pin p. 158 |
: Semigroup whose regular D are semigroup |
|
| Pin pp. 154, 155, 158 |
: Semigroup whose regular D are aperiodic semigroup |
|
| Pin p. 156, 158 |
/: Lefty trivial semigroup |
|
| Pin pp. 149, 158 |
/: Right trivial semigroup |
|
| Pin pp. 149, 158 |
: Locally trivial semigroup |
|
| Pin pp. 150, 158 |
: Locally groups |
|
| Pin pp. 151, 158 |
Terminology | Defining property | Variety | Reference(s) |
---|---|---|---|
Ordered semigroup |
|
| Pin p. 14 |
|
| Pin pp. 157, 158 | |
|
| Pin pp. 157, 158 | |
|
| Pin pp. 157, 158 | |
|
| Pin pp. 157, 158 | |
locally positive J-trivial semigroup |
|
| Pin pp. 157, 158 |
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have an identity element.
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
In mathematics, an algebraic structure consists of a nonempty set A, a collection of operations on A, and a finite set of identities, known as axioms, that these operations must satisfy.
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed.
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum and a unique infimum. An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor.
In mathematics, a join-semilattice is a partially ordered set that 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. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories.
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In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1. Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under some operations satisfying certain equations.
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers.
Algebra is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics.
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Presently, the term "abstract algebra" is typically used for naming courses in mathematical education, and is rarely used in advanced mathematics.
In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group: uniqueness, double application "cancelling itself out", and the same interaction law with the binary operation as in the case of the group inverse. It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups.
In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements:
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 on the underlying set of a semigroup being non-empty. One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function from S × S to S. This operation vacuously satisfies the closure and associativity axioms of a semigroup. Not excluding the empty semigroup simplifies certain results on semigroups. For example, the result that the intersection of two subsemigroups of a semigroup T is a subsemigroup of T becomes valid even when the intersection is empty.
In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topological notions. Varieties of finite monoids, varieties of finite ordered semigroups and varieties of finite ordered monoids are defined similarly.
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[C&P II] | A. H. Clifford, G. B. Preston (1967). The Algebraic Theory of Semigroups Vol. II (Second Edition). American Mathematical Society. ISBN 0-8218-0272-0 | |
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[Tvm] | Proceedings of the International Symposium on Theory of Regular Semigroups and Applications, University of Kerala, Thiruvananthapuram, India, 1986 | |
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