Wiktionary The term regular can mean normal or obeying rules. Regular may refer to In organizations Regular Army for military usage Regular clergy , members of a religious order subject to a rule of life Regular Force for usage in the Canadian Forces Regular Masonic jurisdictions , or regularity , refers ... mutual recognition. In mathematics, geometry, and statistics Regular cardinal , a cardinal number that is equal to its cofinality Regular category , a kind of category that has similarities to both Abelian categories and to the category of sets Regular code , an algebraic code with a uniform distribution of distances between codewords Regular element disambiguation , certain kinds of elements of an algebraic structure Regular graph , a graph such that all the degrees of the vertices are equal Regular language , a formal language recognizable by a finite state automaton Regular polygon , a polygon where all angles and all sides are equal Regular polyhedron , a 3 dimensional equivalent to a regular polygon Regular prime , a certain kind of prime number Regular representation of a group G, the linear representation afforded by the group action of G on itself Regular ring , a ring such that all ... ideal Regular singular point in theory of ordinary differential equations where the growth of solutions is bounded by an algebraic function Regular space , a topological space in which a point and a closed set can be separated by neighbourhoods Irregularity of a surface Regular surface in algebraic ... infinite chains of sets Castelnuovo Mumford regularity of a coherent sheaf In medicine Regular bowel movements, the opposite of constipation In other uses Protagonist Regular character , a main character who appears more frequently and or prominently than a recurring character Regular expression , a type of pattern describing a set of strings in computer science Regular verb , a grammatical term for a verb with derived forms that are typical for the language Regular customer , a person who ... more details
Regular map may refer to a regular map algebraic geometry , in algebraic geometry, an everywhere defined, polynomial function of algebraic varieties. a regular map graph theory , a symmetric 2 cell embedding of a graph into a closed surface. mathdab ... more details
In geometry a regular 4 polytope can mean either a convex or nonconvex intersecting 4 polytope. See Convex regular 4 polytope There are six convex regular polychora. Schl fli Hess polychoron There are ten star self intersecting regular star polytope star polychora . polychora stub Category Four dimensional geometry Category Polychora ... more details
Regular element may refer to In ring theory , a nonzero element of a ring that is neither a left nor a right zero divisor A regular element of a Lie algebra or Lie group. mathdab ... more details
for the unrelated regular rings introduced by John von Neumann von Neumann regular ring In commutative algebra , a regular ring is a commutative noetherian ring , such that the localization of a ring localization at every prime ideal is a regular local ring that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension . Jean Pierre Serre defines a regular ring as a commutative noetherian ring of finite global homological dimension and shows that this is equivalent to the definition above. For regular rings, Krull dimension agrees with global homological dimension. Examples of regular rings include fields of dimension zero and Dedekind domain s. If A is regular then so is A X , with dimension one greater than that of A . References Jean Pierre Serre , Local algebra , Springer Verlag , 2000, ISBN 3 540 66641 9. Chap.IV.D. Category Ring theory Abstract algebra stub nl Reguliere ring ... more details
In mathematics , especially ring theory , a regular ideal can refer to multiple concepts. In operator ... to be regular or modular if there exists an element e in A such that math ex x in mathfrak i math for every math x in A math . harv Jacobson 1956 In commutative algebra a regular ideal refers to an ideal containing a non zero divisor ref Non zerodivisors in commutative rings are called regular elements . ref . harv Larsen McCarthy 1971 loc p.42 This article will use regular element ideal to help ... a von Neumann regular ideal if for each element x of math mathfrak i math there exists a y in math ..., regular ideal has been used to refer to an ideal J of a ring R such that the quotient ring R J is von Neumann regular ring . ref Burton, D.M. 1970 A first course in rings and ideals. Addison Wesley. Reading, Massachusetts . ref This article will use quotient von Neumann regular to refer to this type of regular ideal. Since the adjective regular has been overloaded, this article adopts the alternative adjectives modular , regular element von Neumann regular , and quotient von Neumann regular to distinguish ... right ideal sfn Jacobson 1956 loc p.6 . Examples In the non unital ring of even integers, 6 is regular ... , then the annihilator of x is a regular maximal right ideal in A . If A is a ring without maximal right ideals, then A cannot have even a single modular right ideal. Regular element ideals Every ring with unity has at least one regular element ideal the trivial ideal R itself. Regular element ideals ... Goldie ring , the converse holds essential ideals are all regular element ideals. harv Lam 1999 loc p.342 Since the product of two regular elements non zerodivisors of a commutative ring R is again a regular element, it is apparent that the product of two regular element ideals is again a regular element ideal. Clearly any ideal containing a regular element ideal is again a regular element ideal. Examples In an integral domain , every nonzero element is a regular element, and so every nonzero ... more details
Regular set may refer to Free regular set Closed regular set regular set disambig Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
Unreferenced date December 2009 In theoretical computer science , a regular grammar is a formal grammar that describes a regular language . Strictly regular grammars A right regular grammar also called ... B where B is in N and denotes the empty string , i.e. the string of length 0. In a left regular ... A where A is in N and is the empty string. An example of a right regular grammar G with N ... cA and S is the start symbol. This grammar describes the same language as the regular expression a bc . A regular grammar is a left or right regular grammar. Some textbooks and articles disallow empty production rules, and assume that the empty string is not present in languages. Extended regular grammars An extended right regular grammar is one in which all rules obey one of B a where B is a non ... and is the empty string. Some authors call this type of grammar a right regular grammar or right linear grammar and the type above a strictly right regular grammar or strictly right linear grammar . An extended left regular grammar is one in which all rules obey one of A a where A is a non ... A is in N and is the empty string. Some authors call this type of grammar a left regular grammar and the type above a strictly left regular grammar . Expressive power There is a direct one to one correspondence between the rules of a strictly left regular grammar and those of a nondeterministic finite ..., the left regular grammars generate exactly all regular language s. The right regular grammars describe the reverses of all such languages, that is, exactly the regular languages as well. Every strict right regular grammar is extended right regular, while every extended right regular grammar can be made ... right regular grammars generate the regular languages as well. Analogously, so do the extended left regular grammars. If empty productions are disallowed, only all regular languages that do not include the empty string can be generated. Mixing left and right regular rules If mixing of left regular ... more details
Separation axiom In topology and related fields of mathematics , a topological space X is called a regular ... 3 sub space usually means a regular Hausdorff space . These conditions are examples of separation axiom s. Definitions Image Regular space.svg 250px thumb right The point x , represented by a dot ... around the open disk V , yet U and V do not touch each other. A topological space X is a regular .... A T sub 3 sub space or regular Hausdorff space is a topological space that is both regular ... if it is both regular and T sub 0 sub . A T sub 0 sub or Kolmogorov space is a topological space ... is Hausdorff then it is T sub 0 sub , and each T sub 0 sub regular space is Hausdorff given two distinct ... presented here for regular and T sub 3 sub are not uncommon, there is significant variation in the literature some authors switch the definitions of regular and T sub 3 sub as they are used here, or use both terms interchangeably. In this article, we will use the term regular freely, but we will usually say regular Hausdorff , which is unambiguous, instead of the less precise T sub 3 sub . For more on this issue, see History of the separation axioms . A locally regular space is a topological space where every point has an open neighbourhood that is regular. Every regular space is locally regular, but the converse is not true. A classical example of a locally regular space that is not regular is the bug eyed line . Relationships to other separation axioms A regular space is necessarily also ... space , a regular space that is also T sub 0 sub must be Hausdorff and thus T sub 3 sub . In fact, a regular Hausdorff space satisfies the slightly stronger condition Urysohn space T sub 2 sub . However ... Hausdorffness all are equivalent in the context of regular spaces. Speaking more theoretically, the conditions of regularity and T sub 3 sub ness are related by Kolmogorov quotient s. A space is regular ... if and only if it s both regular and T sub 0 sub . Thus a regular space encountered in practice can ... more details
In artificial intelligence and operations research , a regular constraint is a kind of global constraint . It can be used to solve a particular type of puzzle called a nonogram or logigrams. External links Paltzer, Nikos. http www.ps.uni sb.de courses seminar ws04 papers paltzer.pdf Regular Language Membership Constraint Category Constraint programming Compu AI stub Mathapplied stub ... more details
inline date October 2011 refimprove date October 2011 POV date October 2011 Baptist Regular Baptists are a diverse group of Baptists in the United States and Canada . The presence of the modifier Regular in their names attests to the strong influence of the early Regular Baptists on the growth of Baptists in North America . Two strains of Baptists emigrated from England to United States America the General ... Association org. 1707 , probably gave rise to the Particulars becoming the Regular Baptists . Early in the 19th century, the two dominant groups of Baptists in the United States Regular Baptists & Separate ... Baptist s. In spite of this, the term Regular Baptist has persisted to this day. List of Regular Baptists ... of Faith London Baptist Confession of Faith General Association of Regular Baptist Churches organized ... now American Baptist Church American Baptist Churches in the USA . Association of Regular Baptist ... in the Midwest, use the name Regular Baptist instead of, or in addition to the name Primitive Baptist . Old Regular Baptist s a primarily Appalachian group of churches achieving separate status late in the 19th century. Union Baptists a strand of Regular Baptists that owes its origin to the American ... distinction between Union Baptists and Regular Baptists. Three associations Original Mountain Union ... may still be in existence. Regular Baptists found in 5 local associations much like the Old Regular ... and bordering areas 4288 members in 63 churches in 1999 . Regular, Old Regular, Primitive ... Baptist Churches in Canada although the FEBCC is not generally considered Regular Baptist , some churches of this Fellowship still carry Regular Baptist as part of their name, especially in British ... Regular Baptist . fatt Sources Association Minutes Giving Glory to God in Appalachia , by Howard Dorgan Encyclopedia of Religion in the South , Samuel S. Hill, editor History of Regular Baptist and Their Ancestors ... Baptist Category Calvinism pt Igreja Batista Regular ... more details
Unreferenced date December 2009 In mathematics , the regular part of a Laurent series consists of the series of terms with positive powers. That is, if math f z sum n infty infty a n z c n, math then the regular part of this Laurent series is math sum n 0 infty a n z c n. math In contrast, the series of terms with negative powers is the principal part . DEFAULTSORT Regular Part Category Complex analysis ... more details
language theory , a regular language is a formal language that can be expressed using a regular expression . Note that the regular expression features provided with many programming languages are Regular expression Patterns for non regular languages augmented with features that make them capable of recognizing languages that can not be expressed by the formal regular expressions as formally defined below . In the Chomsky hierarchy , regular languages are defined to be the languages that are generated by Type 3 grammars regular grammar s . Regular languages are very useful in input parsing and programming language design. Formal definition The collection of regular languages over an alphabet is defined recursively as follows The empty language is a regular language. For each a a belongs to , the Singleton mathematics singleton language a is a regular language. If A and B are regular languages, then A B union , A B concatenation , and A Kleene star are regular languages. No other languages over are regular. See Regular expression Formal language theory regular expression for its syntax and semantics. Note that the above cases are in effect the defining rules of regular expression. Examples All finite languages are regular in particular the empty string language is regular. Other typical examples include the language consisting of all strings over the alphabet ... several a s followed by several b s. A simple example of a language that is not regular is the set of strings ... A regular language satisfies the following equivalent properties it can be accepted by a nondeterministic ... finite automaton , or the more general alternating finite automaton it can be generated by a regular ... definition of regular languages. Complexity results In computational complexity theory , the complexity class of all regular languages is sometimes referred to as REGULAR or REG and equals DSPACE O 1 ... size . REGULAR AC0 AC sup 0 sup , since it trivially contains the parity problem of determining ... more details
unreferenced date October 2008 A regular army consists of the permanent force of a country s army that is maintained under arms during peacetime. Countries that use the term include Australian Army British Army Formation and structure British Army Canadian Forces , specifically Regular Force Egyptian army Indian Army Nepalese Army New Zealand Army Regular Army New Zealand Army Singapore Army Sri Lanka Army Regular Army United States United States Army Chinese Army Pakistan Army Pakistan Army 12th Regular Regiment See also Military reserve force Standing army Irregulars Category Types of military forces mil unit stub de Konstitutionelle Armee tr D zenli ordu zh ... more details
In astronomy, a regular moon is a natural satellite following a relatively close and generally prograde orbit with little orbital inclination or orbital eccentricity eccentricity . They are believed to have formed in orbit about their primary astronomy primary , as opposed to irregular moon s, which were captured. There are at least 55 regular satellites of the eight planets one at Earth, eight at Jupiter, 22 named regular moons at Saturn not counting hundreds or thousands of moonlet s , 18 known at Uranus, and 6 small regular moons at Neptune. Large Triton appears to have been captured. It is thought that Pluto s four moons and Haumea s two were formed in orbit about those dwarf planet s out of Collisional family debris created in giant collisions . See also Irregular moon Inner moon Category Moons astronomy stub ca Sat l lit regular eo Regula satelito zh ... more details
Graph families defined by their automorphisms In graph theory , a regular graph is a graph mathematics ... theory degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree ... 29 isbn 9789810218591 ref A regular graph with vertices of degree span class texhtml var k var span is called a span class texhtml var k var span regular graph or regular graph of degree span class texhtml var k var span . Regular graphs of degree at most 2 are easy to classify A 0 regular graph consists of disconnected vertices, a 1 regular graph consists of disconnected edges, and a 2 regular graph consists of disconnected cycle graph theory cycle s. A 3 regular graph is known as a cubic graph . A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number ... in common. The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices. The complete graph math K m math is strongly regular for any math ... var k var span regular graph on span class texhtml 2 var k var 1 span vertices has a Hamiltonian cycle . gallery Image 0 regular graph.svg 0 regular graph Image 1 regular graph.svg 1 regular graph Image 2 regular graph.svg 2 regular graph Image 3 regular graph.svg 3 regular graph gallery Algebraic properties Let A be the adjacency matrix of a graph. Then the graph is regular if and only ...,v n math , we have math sum i 1 n v i 0 math . A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one. ref name Cvetkovic There is also a criterion for regular and connected graphs a graph is connected and regular if and only if the matrix J , with math J ij 1 ... needed date February 2009 Let G be a k regular graph with diameter D and eigenvalues of adjacency ... date March 2009 Generation Regular graphs may be generated by the GenReg program. ref cite journal last Meringer first Markus year 1999 title Fast generation of regular graphs and construction of cages ... more details
In field theory mathematics field theory , a branch of algebra, a field extension math L k math is said to be regular if k is algebraically closed in L and L is separable extension separable over k , or equivalently, math L otimes k overline k math is an integral domain when math overline k math is the algebraic closure of math k math that is, to say, math L, overline k math are linearly disjoint over k . In particular, any field extension of an algebraically closed field is regular. Also, a purely transcendental extension of a field is regular. There is also a similar notion a field extension math L k math is said to be self regular if math L otimes k L math is an integral domain. A self regular extension is algebraically closed in k . However, a self regular extension is not necessarily regular. Citation needed date February 2010 References M. Nagata 1985 . Commutative field theory new edition, Shokado. Japanese http www.shokabo.co.jp mybooks ISBN978 4 7853 1309 8.htm P.M. Cohn 2003 . Basic algebra. A. Weil, Foundations of algebraic geometry . Category Field theory Abstract algebra stub ... more details
2 Set of convex regular p gons align center colspan 2 Image Triangle.Equilateral.svg 50px Image ... 50px Image Enneagon.svg 50px Image Decagon.svg 50px br Regular polygons bgcolor e7dcc3 Edge ... , isotoxal figure isotoxal A regular polygon is a polygon that is Equiangular polygon equiangular all angles are equal in measure and equilateral all sides have the same length . Regular polygons ... apply to all regular polygons, whether convex or star polygon star . A regular n sided polygon has rotational symmetry of order n . All vertices of a regular polygon lie on a common circle the circumscribed ..., this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the mid point. A regular n sided polygon can be constructed with compass and straightedge ... constructible polygon . Symmetry The symmetry group of an n sided regular polygon is dihedral group ... then all axes pass through a vertex and the midpoint of the opposite side. Regular convex polygons All regular simple polygon s a simple polygon is one that does not intersect itself anywhere are convex. Those having the same number of sides are also Similarity geometry similar . An n sided convex regular ... align center File Regular polygon 3.svg 50px BR Equilateral triangle Equilateral BR triangle BR 3 File Regular polygon 4.svg 50px BR square geometry Square BR 4 File Regular polygon 5.svg 50px BR Pentagon BR 5 File Regular polygon 6.svg 50px BR Hexagon BR 6 File Regular polygon 7.svg 50px BR Heptagon BR 7 File Regular polygon 8.svg 50px BR Octagon BR 8 File Regular polygon 9.svg 50px BR Enneagon BR 9 File Regular polygon 10.svg 50px BR Decagon BR 10 align center File Regular polygon 11.svg 50px BR Hendecagon BR 11 File Regular polygon 12.svg 50px BR Dodecagon BR 12 File Regular polygon 13.svg 50px BR Tridecagon BR 13 File Regular polygon 14.svg 50px BR Tetradecagon BR 14 File Regular polygon 15.svg 50px BR Pentadecagon BR 15 File Regular polygon 16.svg 50px BR Hexadecagon BR 16 File Regular ... more details
The term Clerks Regular singular Clerk Regular designates a number of Catholic Church Catholic Catholic priesthood priests clerics who are members of a religious order regular of priests, but in the strictest sense of the word are not Canons Regular . Canonical Status By clerks regular are meant those ... Juris Canonici the term clerks regular is often used for canons regular, and regular clerks are classed by authors as a branch or modern adaptation of the once world famous family of regular canons ... s, and friar s. Clerks regular are distinguished from the purely monastic bodies, or monks ... of life. Clerks regular as clerics must retain some appearance of clerical dress distinct from the religious ... of learning, they are not primarily priests. Finally, clerks regular differ from canons regular ... The exact date at which clerks regular appeared in the Church cannot be absolutely determined. Regular ... hold that the clerks regular were founded by Christ Himself. In this opinion the Apostles were the first regular clerks, being constituted by Christ ministers par excellence of His Church and called ... the founder of the regular clerks and canons, and upon Rule of St. Augustine his rule have been built the constitutions of the canons regular and an immense number of the religious communities of the Middle Ages , besides those of the clerks regular established in the sixteenth century. During the whole medieval period the clerks regular were represented by the regular canons who under the name of the Canons Regular or Black Canons of St. Augustine , the Premonstratensians or White Canons alias .... It was not until the sixteenth century that clerks regular in the modern and strictest sense of the word ... life, instituted the several bodies which, under the names of the various orders or regular clerics ... the clerks regular, that their mode of life was chosen as the pattern for all the various communities ... been so prolific. The first order of clerks regular to be founded was the Congregation of Clerks Regular ... more details
In category theory , a regular category is a category with limit category theory finite limits and coequalizer s of kernel pairs, satisfying certain exactness conditions. In that way, regular categories ... additivity. At the same time, regular categories provide a foundation for the study of a fragment of first order logic , known as regular logic. Definition A category C is called regular if it satisfies ... in C , and div style text align center Image Regular category 1.png div is a pullback category ... isomorphism . If f X Y is a morphism in C , and div style text align center Image Regular category 2.png div is a pullback, and if f is a regular epimorphism , then g is a regular epimorphism as well. A regular epimorphism is an epimorphism which appears as a coequalizer of some pair of morphisms. Examples Examples of regular categories include Category of sets Set , the category of Set mathematics ... given by the order relation Abelian categories The following categories are not regular Top , the category ... of small category small categories and functor s Epi mono factorization In a regular category, the regular epimorphism s and the monomorphism s form a factorization system . Every morphism f X Y can be factorized into a regular epimorphism e X E followed by a monomorphism m E Y , so that f me . The factorization is unique in the sense that if e X E is another regular epimorphism and m E Y is another ... and regular functors In a regular category, a diagram of the form math R rightrightarrows X ... regular categories is called regular , if it preserves finite limits and coequalizers of kernel pairs. A functor is regular if and only if it preserves finite limits and exact sequences. For this reason, regular functors are sometimes called exact functors . Functors that preserve finite limits are often said to be left exact . Regular logic and regular categories Regular logic is the fragment ... math , center where math phi math and math psi math are regular Formula mathematical logic formulae ... more details
Portal Mathematics Featured article template class wikitable align right width 320 Regular polytope examples valign top File Regular pentagon.svg 160px BR A regular pentagon is a polygon , a two dimensional ... 160px BR A regular dodecahedron is a polyhedron , a three dimensional polytope, with 12 pentagonal ... 120 cell.png 160px BR A regular dodecaplex is a polychoron , a four dimensional polytope, with 120 ... diagram File Cubic honeycomb.png 160px BR A regular cubic honeycomb is a tessellation , an infinite ... Petrie polygon In mathematics , a regular polytope is a polytope whose symmetry is transitive ..., and are regular polytopes of dimension n . Regular polytopes are the generalized analog in any number of dimensions of regular polygon s for example, the square geometry square or the regular pentagon and regular polyhedra for example, the cube . The strong symmetry of the regular polytopes gives .... Classically, a regular polytope in n dimensions may be defined as having regular Facet geometry facets n &minus 1 faces and regular vertex figure s. These two conditions are sufficient to ensure ... not work for abstract polytope s. A regular polytope can be represented by a Schl fli symbol of the form a, b, c, ...., y, z , with regular facets as a, b, c, ..., y , and regular vertex figures as b, c, ..., y, z . Classification and description Regular polytopes are classified primarily according to their dimensionality. They can be further classified according to symmetry . For example the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and icosahedron . Indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality Simplex Regular simplex Measure polytope Hypercube Cross polytope Orthoplex In two dimensions there are infinitely many regular polygon ... more details
In complex analysis , see holomorphic function . In mathematics , a regular function is a function that is analytic function analytic and single valued unique in a given region. ref http mathworld.wolfram.com RegularFunction.html Wolfram Mathworld Regular Function ref In complex analysis, any complex regular function is known as a holomorphic function . In algebraic geometry the term takes up a more specific definition, referring to an everywhere defined, polynomial function on an algebraic variety V with values in the field mathematics field K over which V is defined. For example, if V is the affine line over K , the regular functions on V make up a commutative ring , under pointwise multiplication of functions, isomorphic with the polynomial ring in one variable over K . In other words, the regular functions are just polynomials in some natural parameter on the affine line. More generally, for any affine variety V , the regular functions make up the coordinate ring of V , often written K V . This can be expressed in other ways. A regular function is the same as a morphism to the affine line, or in the language of scheme theory a global section of the structure sheaf . The reason for looking at regular functions becomes more apparent when one allows V to be a projective variety . Then regular functions on V become rare. For example morphisms from a projective space to the affine line must be constant regular functions on a projective space are constant functions. The same is true for any connected projective variety this can be viewed as an algebraic analogue of Liouville s theorem complex analysis Liouville s theorem in complex analysis . In fact taking the function field of an algebraic variety function field K V of an irreducible variety irreducible algebraic curve V , the functions F in the function field may all be realised as morphisms from V to the projective line ... date December 2009 reflist DEFAULTSORT Regular Function Category Algebraic varieties Category Types ... more details
Regular matrix may refer to regular stochastic matrix , a stochastic matrix such that all the entries of some power of the matrix are positive. e.g. Lewis, Matrix Theory, p. 169 invertible matrix this usage is rare . e.g. Plato et al., Concise Numerical Algebra, p. 60 the opposite of irregular matrix , a matrix with a different number of entries in each row. mathdab ... more details
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 . ref Howie 1995 54. ref Regular semigroups are one of the most ... s relations . ref Howie 2002. ref Origins Regular semigroups were introduced by J. A. Green in his influential ... study of regular semigroups which led Green to define his celebrated Green s relations relations ... ways in which to define a regular semigroup S 1 for each a in S , there is an x in S , which ... way of expressing definition 2 above is to say that in a regular semigroup, V a is nonempty, for every ... aba a . ref Clifford and Preston 1961 p. 26. ref A regular semigroup in which idempotent s commute is an inverse semigroup , that is, every element has a unique inverse. To see this, let S be a regular ... of a is shown to be unique. Conversely, it can be shown that any inverse semigroup is a regular ... 6391 ref Theorem. The homomorphic image of a regular semigroup is regular. ref Howie 1995 Lemma 2.4.4. ref Examples of regular semigroups Every group mathematics group is regular. Every inverse semigroup is regular. Every band mathematics band idempotent semigroup is regular in the sense of this article, though this is not what is meant by a band mathematics Regular bands regular band . The bicyclic semigroup is regular. Any transformation semigroup full transformation semigroup is regular. A Rees matrix semigroup is regular. Green s relations Recall that the principal ideal s of a semigroup .... In a regular semigroup S , however, an element a axa automatically belongs to these ideals, without recourse to adjoining an identity. Green s relations can therefore be redefined for regular ... In a regular semigroup S , every math mathcal L math and math mathcal R math class contains at least .... ref Theorem. Let S be a regular semigroup, and let a and b be elements of S . Then math a , mathcal ... mathcal R math class is unique. ref name Howie 1995 Theorem 5.1.1 Special classes of regular semigroups ... more details
A regular polyhedron is a polyhedron whose faces are Congruence geometry congruent regular polygons which are assembled in the same way around each vertex geometry vertex . A regular polyhedron is highly ... on its Flag geometry flag s. This last alone is a sufficient definition. A regular polyhedron ... and m the number of faces meeting at each vertex. There are 5 finite regular polyhedra, which are called ... pair dodecahedron icosahedron 5,3 . The regular polyhedra There are five Convex polygon convex regular ... regular star polyhedron star polyhedra , the Kepler Poinsot polyhedra Main Kepler Poinsot polyhedra ... . All the dihedral angle s of the polyhedron are equal. All the vertex figure s of the polyhedron are regular ... spheres A regular polyhedron has all of three related spheres other polyhedra lack at least one kind ... to all edges. A circumsphere , tangent to all vertices. Symmetry The regular polyhedra are the most ... solids have an Euler characteristic of 2. Some of the regular stars have a Kepler Poinsot polyhedra The Euler characteristic different value . Duality of the regular polyhedra The regular polyhedra come ... This section is linked from Polyhedron See also Regular polytope History of discovery Regular polytope ... of at least some of the regular polyhedra, as evidenced by the discovery near Padua in Northern ... needed date February 2007 . Greeks The earliest known written records of the regular convex solids ... solids . One might characterise the Greek definition as follows A regular polygon is a Convex polygon convex planar figure with all edges equal and all corners equal A regular polyhedron is a solid convex figure with all faces being congruent regular polygons, the same number arranged all alike around ... are regular, the square base is not congruent to the triangular sides , or the shape formed by joining ... triangles, that is, congruent and regular, some vertices have 3 triangles and others have 4 . This concept of a regular polyhedron would remain unchallenged for almost 2000 years. Regular star polyhedra ... more details
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