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). Bands were first studied and named by A. H. Clifford ( 1954 ).
37-846: (Redirected from Bands ) [REDACTED] Look up band or Band in Wiktionary, the free dictionary. Band or BAND may refer to: Places [ edit ] Bánd , a village in Hungary Band, Iran , a village in Urmia County, West Azerbaijan Province, Iran Band, Mureș , a commune in Romania Band-e Majid Khan , a village in Bukan County, West Azerbaijan Province, Iran People [ edit ] Band (surname) , various people with
74-479: A = a a a {\displaystyle a=aaa} so a a = a a a a = a ( a a ) a = a {\displaystyle aa=aaaa=a(aa)a=a} and also the following implication holds in any semigroup: x y = y x ⟹ ( x y ) ( y x ) = ( y x ) ( x y ) {\displaystyle xy=yx\implies (xy)(yx)=(yx)(xy)} . So this semigroup which
111-429: A lattice , in which the meet of two varieties is their intersection and the join of two varieties is the smallest variety that contains both of them. The complete structure of this lattice is known; in particular, it is countable , complete , and distributive . The sublattice consisting of the 13 varieties of regular bands is shown in the figure. The varieties of left-zero bands, semilattices, and right-zero bands are
148-404: A preorder that may be defined as x ≤ y {\displaystyle x\leq y} if x y = x {\displaystyle xy=x} . Requiring commutativity implies that this preorder becomes a (semilattice) partial order . A left-zero band is a band satisfying the equation whence its Cayley table has constant rows. Symmetrically, a right-zero band
185-450: A 2002 single by Mando Diao Other uses in arts, entertainment and media [ edit ] Band, nickname of Brazilian television network Rede Bandeirantes The Band (film) , a 1978 Israeli film The Band (musical) , a 2017 musical by Tim Firth with the music of Take That Clothing, jewelry, and accessories [ edit ] Armband or arm band Smart band a band with electronic component Microsoft Band ,
222-450: A 2002 single by Mando Diao Other uses in arts, entertainment and media [ edit ] Band, nickname of Brazilian television network Rede Bandeirantes The Band (film) , a 1978 Israeli film The Band (musical) , a 2017 musical by Tim Firth with the music of Take That Clothing, jewelry, and accessories [ edit ] Armband or arm band Smart band a band with electronic component Microsoft Band ,
259-540: A commune in Romania Band-e Majid Khan , a village in Bukan County, West Azerbaijan Province, Iran People [ edit ] Band (surname) , various people with the surname Arts, entertainment, and media [ edit ] Music [ edit ] Musical ensemble , a group of people who perform instrumental or vocal music Band (rock and pop) , a small ensemble that plays rock or pop Concert band , an ensemble of woodwind, brass, and percussion instruments Dansband , band playing popular music for
296-663: A metal ring indicating the wearer is married Military [ edit ] Bands (Italian Army irregulars) , 19th- and 20th-century military units in the service of the Italian "Regio Esercito" Female order of the Band , a medieval military order native to Spain Order of the Band , a medieval military order native to Spain Science and technology [ edit ] Band (algebra) , an idempotent semigroup Band (order theory) ,
333-458: A metal ring indicating the wearer is married Military [ edit ] Bands (Italian Army irregulars) , 19th- and 20th-century military units in the service of the Italian "Regio Esercito" Female order of the Band , a medieval military order native to Spain Order of the Band , a medieval military order native to Spain Science and technology [ edit ] Band (algebra) , an idempotent semigroup Band (order theory) ,
370-400: A musical ensemble that plays jazz music Marching band , a group of instrumental musicians who generally perform outdoors School band , a group of student musicians who rehearse and perform instrumental music The Band , a Canadian-American rock and roll group The Band (album) , The Band's eponymous 1969 album "Bands" (song) , by American rapper Comethazine "The Band",
407-419: A numbered metal band on a bird's leg for identification Electronic band structure of electrons in solid-state physics Gastric band , a human weight-control measure Signaling (telecommunications) : In-band signaling Out-of-band Birds Are Not Dinosaurs, or BAND, a controversial stance on the origin of birds Society and government [ edit ] Band (First Nations Canada) ,
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#1732765046534444-419: A numbered metal band on a bird's leg for identification Electronic band structure of electrons in solid-state physics Gastric band , a human weight-control measure Signaling (telecommunications) : In-band signaling Out-of-band Birds Are Not Dinosaurs, or BAND, a controversial stance on the origin of birds Society and government [ edit ] Band (First Nations Canada) ,
481-443: A partner-dancing audience Jazz band , a musical ensemble that plays jazz music Marching band , a group of instrumental musicians who generally perform outdoors School band , a group of student musicians who rehearse and perform instrumental music The Band , a Canadian-American rock and roll group The Band (album) , The Band's eponymous 1969 album "Bands" (song) , by American rapper Comethazine "The Band",
518-425: A rectangular band because for any pair ( i , j ) we have ( i , j ) · ( i , j ) = ( i , j ) so every element is idempotent and the first characteristic identity follows from the second together with idempotence. But a magma that satisfies only the identities for the first characteristic and idempotence need not be associative so the second characteristic only follows from
555-446: A smart band with smartwatch features from the software giant Bandolier or bandoleer, an ammunition belt Bands (neckwear) , two pieces of cloth fitted around the neck as part of formal clothing for clergy, academics, and lawyers Belt (clothing) , a flexible band or strap, typically made of leather or heavy cloth, and worn around the waist Strap , an elongated flap or ribbon, usually of fabric or leather Wedding band ,
592-446: A smart band with smartwatch features from the software giant Bandolier or bandoleer, an ammunition belt Bands (neckwear) , two pieces of cloth fitted around the neck as part of formal clothing for clergy, academics, and lawyers Belt (clothing) , a flexible band or strap, typically made of leather or heavy cloth, and worn around the waist Strap , an elongated flap or ribbon, usually of fabric or leather Wedding band ,
629-583: A social grouping of certain animals of the same species Ribbon (disambiguation) Stripe (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Band . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Band&oldid=1230144074 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description
666-583: A social grouping of certain animals of the same species Ribbon (disambiguation) Stripe (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Band . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Band&oldid=1230144074 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description
703-441: A solid subset of an ordered vector space that contains its supremums Band (radio) , a range of frequencies or wavelengths in radio and radar, specifically: Frequency band LTE frequency bands used for cellphone data Shortwave bands UMTS frequency bands used for cellphones BAND (software) , a mobile app that facilitates group communication Band cell , a type of white blood cell Bird banding , placing
740-441: A solid subset of an ordered vector space that contains its supremums Band (radio) , a range of frequencies or wavelengths in radio and radar, specifically: Frequency band LTE frequency bands used for cellphone data Shortwave bands UMTS frequency bands used for cellphones BAND (software) , a mobile app that facilitates group communication Band cell , a type of white blood cell Bird banding , placing
777-424: Is a band S satisfying We can also say a normal band is a band S satisfying This is the same equation used to define medial magmas , so a normal band may also be called a medial band, and normal bands are examples of medial magmas. A left-regular band is a band S satisfying If we take a semigroup and define a ≤ b if ab = b , we obtain a partial ordering if and only if this semigroup
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#1732765046534814-459: Is a band is actually a nowhere commutative semigroup: In any semigroup the first identity also implies the second because xyz = xy ( zxz ) = ( x ( yz ) x ) z = xz . The idempotents of a rectangular semigroup form a sub band that is a rectangular band but a rectangular semigroup may have elements that are not idempotent. In a band the second identity obviously implies the first but that requires idempotence. There exist semigroups that satisfy
851-470: Is a left-regular band. Left-regular bands thus show up naturally in the study of posets . A right-regular band is a band S satisfying Any right-regular band becomes a left-regular band using the opposite product. Indeed, every variety of bands has an 'opposite' version; this gives rise to the reflection symmetry in the figure below. A regular band is a band S satisfying When partially ordered by inclusion, varieties of bands naturally form
888-436: Is different from Wikidata All article disambiguation pages All disambiguation pages band [REDACTED] Look up band or Band in Wiktionary, the free dictionary. Band or BAND may refer to: Places [ edit ] Bánd , a village in Hungary Band, Iran , a village in Urmia County, West Azerbaijan Province, Iran Band, Mureș ,
925-448: Is different from Wikidata All article disambiguation pages All disambiguation pages Band (algebra) The lattice of varieties of bands was described independently in the early 1970s by Biryukov, Fennemore and Gerhard. Semilattices , left-zero bands , right-zero bands , rectangular bands , normal bands , left-regular bands , right-regular bands and regular bands are specific subclasses of bands that lie near
962-422: Is idempotent which means it is a band. Thus in any nowhere commutative semigroup So x {\displaystyle x} commutes with x y x {\displaystyle xyx} and thus x y x = x {\displaystyle xyx=x} which is the first characteristic identity. Now assume that the first identity holds in a semigroup. This identity implies idempotence:
999-807: Is independent of J , and vice versa. This is why the above result only gives an equivalence between nonempty rectangular bands and pairs of nonempty sets. Rectangular bands are also the T -algebras, where T is the monad on Set with T ( X )= X × X , T ( f )= f × f , η X {\displaystyle \eta _{X}} being the diagonal map X → X × X {\displaystyle X\to X\times X} , and μ X ( ( x 11 , x 12 ) , ( x 21 , x 22 ) ) = ( x 11 , x 22 ) {\displaystyle \mu _{X}((x_{11},x_{12}),(x_{21},x_{22}))=(x_{11},x_{22})} . A normal band
1036-711: Is isomorphic to a direct product of a left-zero band and a right-zero band. All rectangular bands of prime order are zero bands, either left or right. A rectangular band is said to be purely rectangular if it is not a left-zero or right-zero band. In categorical language, one can say that the category of nonempty rectangular bands is equivalent to S e t ≠ ∅ × S e t ≠ ∅ {\displaystyle \mathrm {Set} _{\neq \emptyset }\times \mathrm {Set} _{\neq \emptyset }} , where S e t ≠ ∅ {\displaystyle \mathrm {Set} _{\neq \emptyset }}
1073-575: Is one satisfying so that the Cayley table has constant columns. A rectangular band is a band S that satisfies In any semigroup, the first identity is sufficient to characterize a nowhere commutative semigroup , the proof of this follows. Let a semigroup be nowhere commutative. In any flexible magma ( a a ) a = a ( a a ) {\displaystyle (aa)a=a(aa)} so every element commutes with its square. So in any nowhere commutative semigroup every element
1110-431: Is the category with nonempty sets as objects and functions as morphisms. This implies not only that every nonempty rectangular band is isomorphic to one coming from a pair of sets, but also these sets are uniquely determined up to a canonical isomorphism, and all homomorphisms between bands come from pairs of functions between sets. If the set I is empty in the above result, the rectangular band I × J
1147-418: The bottom of this lattice and which are of particular interest; they are briefly described below. A class of bands forms a variety if it is closed under formation of subsemigroups, homomorphic images and direct products . Each variety of bands can be defined by a single defining identity . Semilattices are exactly commutative bands; that is, they are the bands satisfying the equation Bands induce
Band - Misplaced Pages Continue
1184-634: The first in a semigroup. Any rectangular band is isomorphic to one of the above form (either S {\displaystyle S} is empty, or pick any element e ∈ S {\displaystyle e\in S} , and then ( s ↦ ( s e , e s ) {\displaystyle s\mapsto (se,es)} ) defines an isomorphism S ≅ S e × e S {\displaystyle S\cong Se\times eS} ). Left-zero and right-zero bands are rectangular bands, and in fact every rectangular band
1221-955: The primary unit of First Nations Government in Canada Band society , a small group of humans in a simple form of society Tribe (Native American) , a tribe, band, nation, or other group or community of Indigenous peoples in the United States Other uses [ edit ] Rubber band The Band (professional wrestling) , the Total Nonstop Wrestling name for the professional wrestling stable New World Order See also [ edit ] All pages with titles beginning with Band All pages with titles containing Band Band of Brothers (disambiguation) Bandage Banding (disambiguation) Bandy (disambiguation) Bend (disambiguation) Drum and bugle corps (disambiguation) Herd ,
1258-748: The primary unit of First Nations Government in Canada Band society , a small group of humans in a simple form of society Tribe (Native American) , a tribe, band, nation, or other group or community of Indigenous peoples in the United States Other uses [ edit ] Rubber band The Band (professional wrestling) , the Total Nonstop Wrestling name for the professional wrestling stable New World Order See also [ edit ] All pages with titles beginning with Band All pages with titles containing Band Band of Brothers (disambiguation) Bandage Banding (disambiguation) Bandy (disambiguation) Bend (disambiguation) Drum and bugle corps (disambiguation) Herd ,
1295-400: The second characteristic identity above. The two together also imply associativity (xy)z = x(yz) . Any magma that satisfies these two rectangular identities and idempotence is therefore a rectangular band. So any magma that satisfies both the characteristic identities (four separate magma identities) is a band and therefore a rectangular band. The magma operation defined above is
1332-518: The second identity but are not bands and do not satisfy the first. There is a complete classification of rectangular bands. Given arbitrary sets I and J one can define a magma operation on I × J by setting This operation is associative because for any three pairs ( i x , j x ) , ( i y , j y ) , ( i z , j z ) we have These two magma identities (xy)z = xz and x(yz) = xz are together equivalent to
1369-408: The surname Arts, entertainment, and media [ edit ] Music [ edit ] Musical ensemble , a group of people who perform instrumental or vocal music Band (rock and pop) , a small ensemble that plays rock or pop Concert band , an ensemble of woodwind, brass, and percussion instruments Dansband , band playing popular music for a partner-dancing audience Jazz band ,
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