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In mathematics, a regular semigroup is a semigroup S in which every element is regular, i.e., for each element a in S there exists an element x in S such
Regular_semigroup
Algebraic structure
these we mention: regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There are also interesting
Semigroup
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
Completely_regular_semigroup
Families of certain algebraic structures
mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying
Special_classes_of_semigroups
Semigroup in abstract algebra
mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism
Semigroup_with_involution
Generalization of additive and multiplicative inverses
an I-semigroup and a *-semigroup. A class of semigroups important in semigroup theory are completely regular semigroups; these are I-semigroups in which
Inverse_element
In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under
Transformation_semigroup
Structure in group theory (in mathematics)
that x = xyx and y = yxy, i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for
Inverse_semigroup
Topics referred to by the same term
Neumann regular ring, or absolutely flat ring (unrelated to the previous sense) Regular semi-algebraic systems in computer algebra Regular semigroup, related
Regular
Algebraic structure in mathematics
mathematics, the four-spiral semigroup is a special semigroup generated by four idempotent elements. This special semigroup was first studied by Karl Byleen
Four-spiral_semigroup
published in 1979. Every catholic semigroup either is a regular semigroup or has precisely one element that is not regular, much like the partitioners of
Catholic_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
Bicyclic_semigroup
Semigroup in which every element is idempotent
In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square)
Band_(algebra)
Mathematical group
Nambooripad's partial order) is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies. Since the same
Nambooripad_order
Rings admitting weak inverses
Neumann regular rings include π-regular rings, left/right semihereditary rings, left/right nonsingular rings and semiprimitive rings. Regular semigroup Weak
Von_Neumann_regular_ring
regular. A regular semigroup is a semigroup in which every element is regular. This is a stronger notion than weak inverse. Every regular semigroup is
Weak_inverse
orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup. In more recent terminology, an orthodox semigroup is a regular E-semigroup
Orthodox_semigroup
a semigroup. The set of idempotents in a semigroup is a biordered set and every biordered set is the set of idempotents of some semigroup. A regular biordered
Biordered_set
Type of semigroup
quasi-periodic semigroup, group-bound semigroup, completely π-regular semigroup, strongly π-regular semigroup (sπr), or just π-regular semigroup (although
Epigroup
Most widely known generalized inverse of a matrix
In abstract algebra, a Moore–Penrose inverse may be defined on a *-regular semigroup. This abstract definition coincides with the one in linear algebra
Moore–Penrose_inverse
Rees matrix semigroups are a special class of semigroups introduced by David Rees in 1940. They are of fundamental importance in semigroup theory because
Rees_matrix_semigroup
Function that applies a set to itself
on a given base set, together with function composition, forms a regular semigroup. For a finite set of cardinality n, there are nn transformations and
Transformation_(function)
In abstract algebra, an E-dense semigroup (also called an E-inversive semigroup) is a semigroup in which every element a has at least one weak inverse
E-dense_semigroup
more general class, in particular, a regular semigroup that is also an E-semigroup is known as an orthodox semigroup. Weipoltshammer proved that the notion
E-semigroup
Indian mathematician (1935–2020)
mathematician who made fundamental contributions to the structure theory of regular semigroups. Nambooripad was also instrumental in popularising the TeX software
K._S._S._Nambooripad
Group of 𝑛 × 𝑛 invertible matrices
monoid, but occasionally also full linear semigroup, general linear monoid etc. It is actually a regular semigroup. The infinite general linear group or stable
General_linear_group
Algebraic element satisfying some of the criteria of an inverse
mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix A {\displaystyle
Generalized_inverse
Operation on mathematical functions
inverse (called a pseudoinverse) because the symmetric semigroup is a regular semigroup. If Y ⊆ X, then f : X → Y {\displaystyle f:X\to Y} may compose with
Function_composition
Function whose actual domain of definition may be smaller than its apparent domain
{\displaystyle X,} forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on X {\displaystyle
Partial_function
Set that intersects every one of a family of sets
transformation semigroup is a regular semigroup. g {\displaystyle g} acts as a (not necessarily unique) quasi-inverse for f; within semigroup theory this
Transversal_(combinatorics)
Clifford semigroup (sometimes also called "inverse Clifford semigroup") is a completely regular inverse semigroup. It is an inverse semigroup with x x
Clifford_semigroup
Natural number
206 different linear forests on five labeled nodes, and exactly 206 regular semigroups of order four up to isomorphism and anti-isomorphism. Sloane, N. J
206_(number)
Mathematical structure
In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating
Automatic_semigroup
alphabet Σ, or as the induced transformation semigroup of Q. In older books like Clifford and Preston (1967) semigroup actions are called "operands". In category
Semiautomaton
American mathematician (1908–1992)
theory of semigroups. Vol. 2, American Mathematical Society Clifford, Alfred. H. (1974), The Partial Groupoid of Idempotents of a Regular Semigroup, Tulane
Alfred_H._Clifford
Concept in mathematics
and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The study of semigroups as images
Free_monoid
Theorem in convex and algebraic geometry
(this follows from the fact that the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety). The lemma is
Gordan's_lemma
Mathematical conjecture
aperiodic regular digraph can be labeled in this way; their conjecture was proven in 2007 by Avraham Trahtman. A transformation semigroup is synchronizing
Synchronizing_word
Branch of mathematics that studies algebraic structures
lemma Semigroup Subsemigroup Free semigroup Green's relations Inverse semigroup (or inversion semigroup, cf. [1]) Krohn–Rhodes theory Semigroup algebra
List of abstract algebra topics
List_of_abstract_algebra_topics
Theorem of dominion in abstract algebra
American mathematician John R. Isbell in 1966. Dominion is a concept in semigroup theory, within the study of the properties of epimorphisms. For example
Isbell's_zigzag_theorem
Topic in group theory
notion generalizes to semigroups and, as such, is a central construction in the Krohn–Rhodes structure theory of finite semigroups. Let A {\displaystyle
Wreath_product
British mathematician
of RIAS, and they published a book of crystallographic tables. *-regular semigroup Drazin, Charles (25 August 2016). Mapping the Past: A Search for Five
Michael_P._Drazin
Differential operator in mathematics
is a strongly continuous contraction semigroup whose generator is the Laplacian; more generally, the heat semigroup acts contractively on Lp for 1 ≤ p ≤
Laplace_operator
Smallest monoid that recognizes a formal language
ISBN 1-58488-255-7. Zbl 1086.68074. Pin, Jean-Éric (1997). "10. Syntactic semigroups". In Rozenberg, G.; Salomaa, A. (eds.). Handbook of Formal Language Theory
Syntactic_monoid
Mathematical model of computation
automaton SCXML Semiautomaton Semigroup action Sequential logic State diagram Synchronizing word Transformation semigroup Transition system Tree automaton
Finite-state_machine
science such as automata theory, syntactic semigroup, model theory and semigroup theory. The class of regular numerical predicate is denoted C l c a {\displaystyle
Regular_numerical_predicate
Type of topological space in mathematics
on 2015-09-10. Lawson, J.; Madison, B. (1974). "Quotients of k-semigroups". Semigroup Forum. 9: 1–18. doi:10.1007/BF02194829., p. 3 Breuckmann, Tomas;
Locally_compact_space
Proof that every structure with certain properties is isomorphic to another structure
of copies of A. In the study of semigroups, the Wagner–Preston theorem provides a representation of an inverse semigroup S, as a homomorphic image of the
Representation_theorem
Transformations induced by a mathematical group
does not define bijective maps and equivalence relations however. See semigroup action. Instead of actions on sets, we can define actions of groups and
Group_action
Study of abstract machines and automata
automata transformations or as semigroup homomorphisms, when the state space, S, of the automaton is defined as a semigroup Sg. Monoids are also considered
Automata_theory
Hungarian mathematician
attention later turned to semigroups, publishing papers on the decomposition of semigroups and on congruence relations of regular semigroups. His book with Jürgensen
Jenő_Szép
String rewriting system
introduced this notion hoping to solve the word problem for finitely presented semigroups. Only in 1947 was the problem shown to be undecidable— this result was
Semi-Thue_system
Theorem in group theory
similar to the universal embedding theorem, but for semigroups. A semigroup S is a divisor of a semigroup T if it is the image of a subsemigroup of T under
Universal_embedding_theorem
direct product of a right zero semigroup and a group, while a right abelian group is the direct product of a right zero semigroup and an abelian group. Left
Right_group
Set of natural numbers
extended from subsets of the special semigroup of natural numbers with addition to subsets of semigroups and partial semigroups in general. A variant of Hindman's
IP_set
Random process independent of past history
X} and ( P t ) t ≥ 0 {\displaystyle (P_{t})_{t\geq 0}} the transition semigroup of the process. Transition functions are generalizations of the transition
Markov_chain
American mathematician
221–258. Benjamin Steinberg. "A topological approach to inverse and regular semigroups." Pacific Journal of Mathematics, vol. 208 (2003), no. 2, pp. 367–396
John_R._Stallings
Concept in topology
\\G\mapsto F+G\end{cases}}} is continuous. More generally, if S is a semigroup with the discrete topology, the operation of S can be extended to βS,
Stone–Čech_compactification
Class of algebraic structures
a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups. Quasivariety Birkhoff, G. (Oct 1935), "On the
Variety_(universal_algebra)
"An interesting combinatorial method in the theory of locally finite semigroups". Pacific Journal of Mathematics. 36 (2): 285–289. doi:10.2140/pjm.1971
Partition_regularity
Branch of mathematics
specialized structure by adding constraints. For example, a magma becomes a semigroup if its operation is associative. Homomorphisms are tools to examine structural
Algebra
Representation of groups by permutations
original theorem. Wagner–Preston theorem is the analogue for inverse semigroups. Birkhoff's representation theorem, a similar result in order theory Frucht's
Cayley's_theorem
Russian mathematician (1938–2017)
recognizing the solvability of arbitrary equations in free groups and semigroups. At Moscow State University he received his undergraduate degree and in
Gennady_Makanin
Set of integers containing arbitrarily long intervals
of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup ( S , ⋅ ) {\displaystyle (S,\cdot )} and A ⊆ S {\displaystyle
Thick_set
Finite-state machine
monoid is known as the transition monoid, or sometimes the transformation semigroup. The construction can also be reversed: given a δ ^ {\displaystyle {\widehat
Deterministic finite automaton
Deterministic_finite_automaton
Natural number
following 187 and preceding 189. There are 188 different four-element semigroups, and 188 ways a chess queen can move from one corner of a 4 × 4 {\displaystyle
188_(number)
Partial algebra
requirement that there be an identity at each object. Semigroupoids generalise semigroups in the same way that small categories generalise monoids and groupoids
Semigroupoid
algebras. A magma which is both commutative and associative is a commutative semigroup. In the game of rock paper scissors, let M := { r , p , s } {\displaystyle
Commutative_magma
partitions of 12 white objects and 3 black ones 1915 = number of nonisomorphic semigroups of order 5 1916 = sum of first 50 composite numbers 1917 = number of partitions
1000_(number)
(Russian: Свердловская тетрадь) is a collection of unsolved problems in semigroup theory, first published in 1965 and updated every 2 to 4 years since.
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Type of integral domain
series ring K[[X1, ..., Xn]] over a field K (or more generally over a regular UFD such as a PID) is a UFD. On the other hand, the formal power series
Unique_factorization_domain
Jean-Christophe Novelli, and Florent Hivert in 2001. The Chinese monoid has a regular language cross-section a ∗ ( b a ) ∗ b ∗ ( c a ) ∗ ( c b ) ∗ c ∗ ⋯
Chinese_monoid
Branch of mathematical analysis
defined in this way are continuous semigroups with parameter a {\displaystyle a} , of which the original discrete semigroup of { D n ∣ n ∈ Z } {\displaystyle
Fractional_calculus
Algebraic structure with addition, multiplication, and division
modern language, the resulting cyclic Galois group. Gauss deduced that a regular p-gon can be constructed if p = 22k + 1. Building on Lagrange's work, Paolo
Field_(mathematics)
Function that is its own inverse
as (xy)−1 = (y)−1(x)−1. Taken as an axiom, it leads to the notion of semigroup with involution, of which there are natural examples that are not groups
Involution_(mathematics)
Mathematical problem
Postage stamp problem Change-making problem Sylver coinage Numerical semigroup The original source is sometimes incorrectly cited as, in which the author
Coin_problem
Sequence of words formed by specific rules
use this paper as the basis for a 1947 proof "that the word problem for semigroups was recursively insoluble", and later devised the canonical system for
Formal_language
Equation for fixed point of functional composition
group.) The set of hn(x), i.e., of all positive integer iterates of h(x) (semigroup) is called the splinter (or Picard sequence) of h(x). However, all iterates
Schröder's_equation
Fundamental solution to the heat equation, given boundary values
spectral mapping theorem gives a representation of T in the form the semigroup T = e t Δ . {\displaystyle T=e^{t\Delta }.} There are several geometric
Heat_kernel
Type of subset of the natural numbers
set Thick set McLeod, Jillian (2000). "Some Notions of Size in Partial Semigroups" (PDF). Topology Proceedings. 25 (Summer 2000): 317–332. Bergelson, Vitaly
Syndetic_set
Probability concept
complicated in larger matrices. The fact that Q is the generator for a semigroup of matrices P ( t + s ) = e ( t + s ) Q = e t Q e s Q = P ( t ) P ( s
Continuous-time_Markov_chain
Mathematical operation
{T}}.} This property makes the set of all binary relations on a set a semigroup with involution. The composition of (partial) functions (that is, functional
Composition_of_relations
Natural number
a regular nonagon. There are exactly 126 binary strings of length seven that are not repetitions of a shorter string, and 126 different semigroups on
126_(number)
Mathematics term
ab, b → ac, c → db, d → dc followed by the coding a,b → 0, c,d → 1. The regular paperfolding sequence is obtained from the fixed point of the 2-uniform
Morphic_word
are regular band operations. The above symbols D {\displaystyle D} , R {\displaystyle R} and L {\displaystyle L} come, of course, from basic semigroup theory
Skew_lattice
Group of symmetries of an n-dimensional hypercube
Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000), Internat. J. Algebra Comput., 12 (1–2): 85–97
Hyperoctahedral_group
Algebraic structure
An explicit example is the ring of integers Z, a Euclidean domain. All regular local rings are integrally closed as well. A ring whose localizations at
Integrally_closed_domain
other structures. For instance, it generalizes naturally to automatic semigroups. Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio
Automatic_group
Branch of mathematics
structures with a single binary operation are: Magma Quasigroup Monoid Semigroup Group Examples involving several operations include: Ring Field Module
Abstract_algebra
Group in group theory and physics
{\mathcal {L}}=-\sum _{j=1}^{n}(X_{j}^{2}+Y_{j}^{2}),} the corresponding heat semigroup is generated by − 1 2 L {\displaystyle -{\frac {1}{2}}{\mathcal {L}}}
Heisenberg_group
Algebraic structure with addition and multiplication
perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD. The following is a chain of class inclusions
Ring_(mathematics)
Algebra with unique prime factorization
all fractional ideals endowed with the above product is a commutative semigroup and in fact a monoid: the identity element is the fractional ideal R.
Dedekind_domain
of a group or more generally a semigroup is an undirected graph in which the vertices are elements of the group/semigroup and there is an edge between any
Glossary_of_graph_theory
is in S and no factor of length 2 in w is in F. This corresponds to the regular expression ( R A ∗ ∩ A ∗ S ) ∖ A ∗ F A ∗ . {\displaystyle (RA^{*}\cap
Local language (formal language)
Local_language_(formal_language)
Algebraic ring that need not have additive negative elements
makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly. These authors often use rig for
Semiring
Type of group in abstract algebra
group Symmetry in quantum mechanics § Exchange symmetry Symmetric inverse semigroup Symmetric power Jacobson 2009, p. 31 Jacobson 2009, p. 32 Theorem 1.1
Symmetric_group
Generalization of vector spaces from fields to rings
a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r =
Module_(mathematics)
Variation of a finite automaton that runs on infinite input
Perrin, Dominique; Pin, Jean-Éric (2004), Infinite Words: Automata, Semigroups, Logic and Games, Elsevier, ISBN 978-0-12-532111-2 Thomas, Wolfgang (1990)
Ω-automaton
Undecidable decision problem introduced by Emil Post
is undecidable and equivalent to the following Group Problem: is the semigroup generated by a finite set of pairs of words (over a group alphabet) a
Post_correspondence_problem
Commutative ring with no zero divisors other than zero
with this property are called regular, so it is equivalent to require that every nonzero element of the ring be regular. An integral domain is a ring
Integral_domain
REGULAR SEMIGROUP
REGULAR SEMIGROUP
Boy/Male
Indian, Sanskrit
Connector; Regulator
Boy/Male
Shakespearean
King Henry IV, Part 1 and 2' An irregular humorist.
Male
Scandinavian
Scandinavian form of German Reginar, RAGNAR means "wise warrior."
Girl/Female
Muslim
One who remembers Allah regularly
Male
Spanish
Spanish form of Roman Latin Regulus, RÉGULO means "ruler."
Girl/Female
Hebrew
Precious.
Boy/Male
Shakespearean
King Henry IV, Part 1 and 2' Edward Poins, an irregular humorist.
Surname or Lastname
English, of Welsh origin
English, of Welsh origin : variant of Bowen, with the addition of the regular English patronymic suffix -s.Altered spelling of Dutch Bouwens, a variant of Bauwens.
Male
Italian
Italian form of German Reginar, RANIERO means "wise warrior."
Surname or Lastname
North German
North German : variant of Asch.English : variant spelling of Ash (asche was the regular Middle English spelling of this word).
Girl/Female
Arabic, Muslim
Pilgrimage to Makkah Other than Regular Hajj Days
Girl/Female
Indian
One who remembers Allah regularly
Surname or Lastname
English, of Welsh origin
English, of Welsh origin : variant of Bevan, with the addition of the regular English patronymic suffix -s.
Boy/Male
Gujarati, Haryanvi, Hindu, Indian, Kannada, Marathi, Telugu
Regular; Ethical; Good in Nature
Male
German
A derivative of German Reginar, RAINER means "wise warrior."
Girl/Female
Muslim/Islamic
One who remembers Allah regularly
Surname or Lastname
English (Devon)
English (Devon) : unexplained. Possibly an irregular variant of Birchall.
Boy/Male
Hindu, Indian, Tamil
Regular Winner
Surname or Lastname
English
English : nickname probably for a tenant whose feudal obligations included a regular payment in cash or kind (for example bread or salt) of a halfpenny.
Boy/Male
Hindu, Indian, Traditional
Conduct; Regular Performance of Worship
REGULAR SEMIGROUP
REGULAR SEMIGROUP
Girl/Female
Indian, Sanskrit
Eternal
Boy/Male
Indian
Beautiful
Girl/Female
Hindu
Enabling to crossover
Girl/Female
Indian, Punjabi, Sikh
Destroyer of Tyranny
Girl/Female
Hindu, Indian, Malayalam, Marathi
Girl Born in Month of Chaitra
Girl/Female
Hindu
Speech, Goddess Saraswati
Girl/Female
American, British, Christian, Danish, English, French, German, Indian, Jamaican
Cyneburg's Field; Royal Fortress Meadow
Girl/Female
Indian
Definition
Boy/Male
Arabic
Ebony
Biblical
rejoicing; mercy; linen
REGULAR SEMIGROUP
REGULAR SEMIGROUP
REGULAR SEMIGROUP
REGULAR SEMIGROUP
REGULAR SEMIGROUP
pl.
of Tegula
a.
Of or pertaining to the jugular vein; as, the jugular foramen.
a.
Of or pertaining to a tile; resembling a tile, or arranged like tiles; consisting of tiles; as, a tegular pavement.
a.
Thorough; complete; unmitigated; as, a regular humbug.
a.
Governed by rule or rules; steady or uniform in course, practice, or occurence; not subject to unexplained or irrational variation; returning at stated intervals; steadily pursued; orderlly; methodical; as, the regular succession of day and night; regular habits.
n. pl.
A division of Echini which includes the circular, or regular, sea urchins.
v. t.
To cause to become regular; to regulate.
a.
Not regular; not conforming to a law, method, or usage recognized as the general rule; not according to common form; not conformable to nature, to the rules of moral rectitude, or to established principles; not normal; unnatural; immethodical; unsymmetrical; erratic; no straight; not uniform; as, an irregular line; an irregular figure; an irregular verse; an irregular physician; an irregular proceeding; irregular motion; irregular conduct, etc. Cf. Regular.
a.
Measured by an angle; as, angular distance.
adv.
In a regular manner; in uniform order; methodically; in due order or time.
a.
Conformed to a rule; agreeable to an established rule, law, principle, or type, or to established customary forms; normal; symmetrical; as, a regular verse in poetry; a regular piece of music; a regular verb; regular practice of law or medicine; a regular building.
a.
Having all the parts of the same kind alike in size and shape; as, a regular flower; a regular sea urchin.
pl.
of Regulus
a.
Belonging to a monastic order or community; as, regular clergy, in distinction dfrom the secular clergy.
a.
Not regular; not bound by monastic vows or rules; not confined to a monastery, or subject to the rules of a religious community; as, a secular priest.
a.
Irregular in position; having no regular order; as, scattered leaves.
a.
Constituted, selected, or conducted in conformity with established usages, rules, or discipline; duly authorized; permanently organized; as, a regular meeting; a regular physican; a regular nomination; regular troops.
n.
A secular ecclesiastic, or one not bound by monastic rules.
n.
One who is not regular; especially, a soldier not in regular service.
a.
Fig.: Lean; lank; raw-boned; ungraceful; sharp and stiff in character; as, remarkably angular in his habits and appearance; an angular female.