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57-464: The Tafawa Balewa Square (TBS) is a 14.5-hectare (35.8-acre) ceremonial ground (originally called "Race Course") in Lagos Island , Lagos . Lagos Race Course now TBS, was a sports field that hosted horse racing, but included a section for football and ground to play cricket. The land was provided to colonial authorities by Oba Dosunmu in 1859, who thereafter built up the surrounding areas. The course

114-459: A S b ⊆ S a ∧ b . If f is onto, the semilattice L is isomorphic to the quotient of S by the equivalence relation ~ such that x ~ y if and only if f ( x ) = f ( y ) . This equivalence relation is a semigroup congruence, as defined above. Whenever we take the quotient of a commutative semigroup by a congruence, we get another commutative semigroup. The structure theorem says that for any commutative semigroup S , there

171-399: A commutative semigroup, when it exists, is a group. Green's relations , a set of five equivalence relations that characterise the elements in terms of the principal ideals they generate, are important tools for analysing the ideals of a semigroup and related notions of structure. The subset with the property that every element commutes with any other element of the semigroup is called

228-409: A , b ∈ L has a greatest lower bound , denoted a ∧ b . The operation ∧ makes L into a semigroup that satisfies the additional idempotence law a ∧ a = a . Given a homomorphism f  : S → L from an arbitrary semigroup to a semilattice, each inverse image S a = f { a } is a (possibly empty) semigroup. Moreover, S becomes graded by L , in the sense that S

285-439: A binary operation ∘ on the congruence classes: Because ~ is a congruence, the set of all congruence classes of ~ forms a semigroup with ∘, called the quotient semigroup or factor semigroup , and denoted S / ~ . The mapping x ↦ [ x ] ~ is a semigroup homomorphism, called the quotient map , canonical surjection or projection ; if S is a monoid then quotient semigroup is a monoid with identity [1] ~ . Conversely,

342-636: A large protected harbour on the coast of Africa, the island is home to the Yoruba fishing village of Eko , which grew into the modern city of Lagos. The city has now spread out to cover the neighboring islands as well as the adjoining mainland. Lagos Island is connected to the mainland by three large bridges (the Carter Bridge , the Eko Bridge and the Third Mainland Bridge ) which cross Lagos Lagoon to

399-451: A maximal element. By Zorn's lemma , this is equivalent to saying that the ascending chain condition holds: there is no infinite strictly ascending chain of congruences on S . Every ideal I of a semigroup induces a factor semigroup, the Rees factor semigroup , via the congruence ρ defined by x ρ y if either x = y , or both x and y are in I . The following notions introduce

456-485: A meaning must be given to the exponential of tA . As a function of t , exp( tA ) is a semigroup of operators from X to itself, taking the initial state u 0 at time t = 0 to the state u ( t ) = exp( tA ) u 0 at time t . The operator A is said to be the infinitesimal generator of the semigroup. The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as groups or rings . A number of sources attribute

513-408: A monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused with quasigroups , which are generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups preserve from groups the notion of division . Division in semigroups (or in monoids)

570-426: A monoid. A natural example is strings with concatenation as the binary operation, and the empty string as the identity element. Restricting to non-empty strings gives an example of a semigroup that is not a monoid. Positive integers with addition form a commutative semigroup that is not a monoid, whereas the non-negative integers do form a monoid. A semigroup without an identity element can be easily turned into

627-443: A semigroup The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication ): x ⋅ y , or simply xy , denotes the result of applying the semigroup operation to the ordered pair ( x , y ) . Associativity is formally expressed as that ( x ⋅ y ) ⋅ z = x ⋅ ( y ⋅ z ) for all x , y and z in the semigroup. Semigroups may be considered

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684-412: A semigroup S (or more generally, magma ) is an element e such that for all x in S , e ⋅ x = x . Similarly, a right identity is an element f such that for all x in S , x ⋅ f = x . Left and right identities are both called one-sided identities . A semigroup may have one or more left identities but no right identity, and vice versa. A two-sided identity (or just identity )

741-405: A set of two elements {a, b }, eight form semigroups whereas only four of these are monoids and only two form groups. For more on the structure of finite semigroups, see Krohn–Rhodes theory . There is a structure theorem for commutative semigroups in terms of semilattices . A semilattice (or more precisely a meet-semilattice) ( L , ≤) is a partially ordered set where every pair of elements

798-419: A special case of magmas , where the operation is associative, or as a generalization of groups , without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutative , so x ⋅ y is not necessarily equal to y ⋅ x ; a well-known example of an operation that is associative but non-commutative is matrix multiplication . If

855-428: Is a monoid homomorphism . But there are semigroup homomorphisms that are not monoid homomorphisms, e.g. the canonical embedding of a semigroup S without identity into S . Conditions characterizing monoid homomorphisms are discussed further. Let f  : S 0 → S 1 be a semigroup homomorphism. The image of f is also a semigroup. If S 0 is a monoid with an identity element e 0 , then f ( e 0 )

912-510: Is a finest congruence ~ such that the quotient of S by this equivalence relation is a semilattice. Denoting this semilattice by L , we get a homomorphism f from S onto L . As mentioned, S becomes graded by this semilattice. Furthermore, the components S a are all Archimedean semigroups . An Archimedean semigroup is one where given any pair of elements x , y , there exists an element z and n > 0 such that x = yz . The Archimedean property follows immediately from

969-443: Is also a group is called a subgroup . There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotent e of the semigroup there is a unique maximal subgroup containing e . Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here

1026-404: Is an element that is both a left and right identity. Semigroups with a two-sided identity are called monoids . A semigroup may have at most one two-sided identity. If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup. If a semigroup has both a left identity and a right identity, then it has a two-sided identity (which is therefore

1083-426: Is an embedding. This need not always be the case: for example, take S to be the semigroup of subsets of some set X with set-theoretic intersection as the binary operation (this is an example of a semilattice). Since A . A = A holds for all elements of S , this must be true for all generators of G ( S ) as well, which is therefore the trivial group . It is clearly necessary for embeddability that S have

1140-539: Is bounded by Awolowo road, Cable Street, Force road, Catholic Mission street and the 26-storey independence building. The entrance to the square has gigantic sculptures of four white horses hovering above the gate and seven red eagles, which are symbols from the national emblem signifying Strength and Dignity respectively. Other monuments in the square include the Remembrance Arcade (with memorials to World War I , World War II and Nigerian civil war victims) and

1197-412: Is called a zero . Analogous to the above construction, for every semigroup S , one can define S , a semigroup with 0 that embeds S . The semigroup operation induces an operation on the collection of its subsets: given subsets A and B of a semigroup S , their product A · B , written commonly as AB , is the set { ab | a in A and b in B }. (This notion is defined identically as it

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1254-464: Is for groups .) In terms of this operation, a subset A is called If A is both a left ideal and a right ideal then it is called an ideal (or a two-sided ideal ). If S is a semigroup, then the intersection of any collection of subsemigroups of S is also a subsemigroup of S . So the subsemigroups of S form a complete lattice . An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of

1311-505: Is independent of time. There are numerous special classes of semigroups , semigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention: regular semigroups , orthodox semigroups , semigroups with involution , inverse semigroups and cancellative semigroups . There are also interesting classes of semigroups that do not contain any groups except

1368-421: Is not necessarily the case that there are a quotient of S . Both of those relations are transitive. For any subset A of S there is a smallest subsemigroup T of S that contains A , and we say that A generates T . A single element x of S generates the subsemigroup { x | n ∈ Z } . If this is finite, then x is said to be of finite order , otherwise it is of infinite order . A semigroup

1425-710: Is not possible in general. The formal study of semigroups began in the early 20th century. Early results include a Cayley theorem for semigroups realizing any semigroup as a transformation semigroup , in which arbitrary functions replace the role of bijections in group theory. A deep result in the classification of finite semigroups is Krohn–Rhodes theory , analogous to the Jordan–Hölder decomposition for finite groups. Some other techniques for studying semigroups, like Green's relations , do not resemble anything in group theory. The theory of finite semigroups has been of particular importance in theoretical computer science since

1482-450: Is noteworthy as the venue of the call for positive action by zikists in November 1948. The hall was built by trustees selected by Mr Thomas Jones who died in 1913. In his will, he bequeathed the land and funds for a hall and library in his memorial. Freedom Park is becoming a major tourist attraction and it is located on Lagos Island. The park was formerly a prison yard, back when

1539-657: Is one of the Nigerian banks with its head office in Marina, Lagos Island. Another bank that has its head office situated in Lagos Island is the United Bank for Africa (UBA). Other medium and large-scale businesses such as real estate consultancy firms , electrical appliances manufacturers and retail stores are based in Marina, Lagos Island. Located on Nnamdi Azikiwe Street, Idumota , formerly Victoria St, Tom Jones Memorial hall

1596-461: Is said to be periodic if all of its elements are of finite order. A semigroup generated by a single element is said to be monogenic (or cyclic ). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive integers with the operation of addition. If it is finite and nonempty, then it must contain at least one idempotent . It follows that every nonempty periodic semigroup has at least one idempotent. A subsemigroup that

1653-452: Is the identity element in the image of f . If S 1 is also a monoid with an identity element e 1 and e 1 belongs to the image of f , then f ( e 0 ) = e 1 , i.e. f is a monoid homomorphism. Particularly, if f is surjective , then it is a monoid homomorphism. Two semigroups S and T are said to be isomorphic if there exists a bijective semigroup homomorphism f  : S → T . Isomorphic semigroups have

1710-580: Is the principal and central Local Government Area (LGA) in Lagos , Nigeria. It was the capital of Lagos State until 1957. It is part of the Lagos Division. As of the preliminary 2006 Nigerian census , the LGA had a population of 209,437 within an area of just 8.7 km . The LGA only covers the western half of Lagos Island; the eastern half is simply referred to as Lagos Island East LCDA. Lying on Lagos Lagoon ,

1767-477: The center of the semigroup. The center of a semigroup is actually a subsemigroup. A semigroup homomorphism is a function that preserves semigroup structure. A function f  : S → T between two semigroups is a homomorphism if the equation holds for all elements a , b in S , i.e. the result is the same when performing the semigroup operation after or before applying the map f . A semigroup homomorphism between monoids preserves identity if it

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1824-490: The cancellation property . When S is commutative this condition is also sufficient and the Grothendieck group of the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups. Anatoly Maltsev gave necessary and sufficient conditions for embeddability in 1937. Semigroup theory can be used to study some problems in

1881-493: The kernel of any semigroup homomorphism is a semigroup congruence. These results are nothing more than a particularization of the first isomorphism theorem in universal algebra . Congruence classes and factor monoids are the objects of study in string rewriting systems . A nuclear congruence on S is one that is the kernel of an endomorphism of S . A semigroup S satisfies the maximal condition on congruences if any family of congruences on S , ordered by inclusion, has

1938-399: The trivial group ; examples of the latter kind are bands and their commutative subclass – semilattices , which are also ordered algebraic structures . A semigroup is a set S together with a binary operation ⋅ (that is, a function ⋅ : S × S → S ) that satisfies the associative property : More succinctly, a semigroup is an associative magma . A left identity of

1995-407: The 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid . In probability theory , semigroups are associated with Markov processes . In other areas of applied mathematics , semigroups are fundamental models for linear time-invariant systems . In partial differential equations , a semigroup is associated to any equation whose spatial evolution

2052-559: The 26-storey Independence House, built in 1963 which was for a long time, the tallest building in Nigeria. The square has a capacity for 55,000 people. Facilities at the square include a shopping center, airlines ticketing agencies, restaurants, car parking and a bus terminal. The cricket ground, the Tafawa Balewa Square Cricket Oval, is widely considered as the 'traditional home of cricket' in Nigeria. It hosted matches in

2109-700: The Grand Hotel before it was demolished. New Africa House of UAC, Elder Dempster House, Nigerian Ports Authority head office and National Electric Power Authority 's former headquarters are all located along the marina. [REDACTED] Media related to Lagos Island at Wikimedia Commons 6°27′N 3°24′E  /  6.450°N 3.400°E  / 6.450; 3.400 See also: List of schools in Lagos See also: List of hospitals in Lagos See also: List of festivals in Lagos See also: Architecture of Lagos Adjoining an identity to

2166-670: The North-Western sub region of the 2018–19 ICC T20 World Cup Africa Qualifier tournament. The ground was closed for 18 months to complete a renovation from a concrete surface to 10-strip turf to meet the ICC standards. The renovation was completed in January 2022, following which the ground hosted its first international cricket matches in the 2022 Nigeria Invitational Women's T20I Tournament . Major national events at TBS includes Nigeria’s independence celebration which took place on 1 October 1960 with

2223-471: The Prime Minister, Tafawa Balewa , delivering his speech. Democracy Day, as well as other multifarious events such as musical jamborees and religious gatherings. The TBS which is a ceremonial ground in Nigeria have been faced with several environmental issues in the past years. The surrounding fence has been a dumpsite where refuse and other plastic are being deposited. There is also the issue of erosion in

2280-536: The conditions of the soil , the foundations of most of the tall buildings are either piled or raft. Buildings along the marina include National House now occupied by Shell and it is the first tall office building at Marina. The former Central Bank headquarters and the Investment House, headquarters of Bank of Industry were both built-in 1960. The land on which the Investment House was built previously housed

2337-426: The corresponding generator. This has a universal property for morphisms from S to a group: given any group H and any semigroup homomorphism k  : S → H , there exists a unique group homomorphism f  : G → H with k = fj . We may think of G as the "most general" group that contains a homomorphic image of S . An important question is to characterize those semigroups for which this map

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2394-495: The country was still under colonial rule and it was known then as Her Majesty's Broad Street Prisons. Freedom park was created to serve as a national memorial in remembrance of the nation's founding fathers who struggled against colonial rule and fought for the country's independence . The park was opened in 2010 to celebrate Nigeria's 50th independence celebration. The park is now a tourist attraction to both locals and foreigners and you can actually find historical statues all over

2451-512: The district of Ebute Metta . It is also linked to the neighboring island of Ikoyi and to Victoria Island . The Lagos harbor district of Apapa faces the western side of the island. Forming the main commercial district of Lagos, Lagos Island plays host to the main government buildings, shops and offices. The Catholic and Anglican Cathedrals , as well as the Central Mosque, are located here. Historically, Lagos Island (Isale Eko)

2508-468: The field of partial differential equations . Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as an ordinary differential equation on a function space. For example, consider the following initial/boundary value problem for the heat equation on the spatial interval (0, 1) ⊂ R and times t ≥ 0 : Let X = L ((0, 1) R ) be the L space of square-integrable real-valued functions with domain

2565-650: The first use of the term (in French) to J.-A. de Séguier in Élements de la Théorie des Groupes Abstraits (Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton's Theory of Groups of Finite Order . Anton Sushkevich obtained the first non-trivial results about semigroups. His 1928 paper "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" ("On finite groups without

2622-472: The idea that a semigroup is contained in another one. A semigroup T is a quotient of a semigroup S if there is a surjective semigroup morphism from S to T . For example, ( Z /2 Z , +) is a quotient of ( Z /4 Z , +) , using the morphism consisting of taking the remainder modulo 2 of an integer. A semigroup T divides a semigroup S , denoted T ≼ S if T is a quotient of a subsemigroup S . In particular, subsemigroups of S divides T , while it

2679-401: The interval (0, 1) and let A be the second-derivative operator with domain where H is a Sobolev space . Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space X : On an heuristic level, the solution to this problem "ought" to be u ( t ) = exp( tA ) u 0 . However, for a rigorous treatment,

2736-474: The ordering in the semilattice L , since with this ordering we have f ( x ) ≤ f ( y ) if and only if x = yz for some z and n > 0 . The group of fractions or group completion of a semigroup S is the group G = G ( S ) generated by the elements of S as generators and all equations xy = z that hold true in S as relations . There is an obvious semigroup homomorphism j  : S → G ( S ) that sends each element of S to

2793-619: The park. There is also an amphitheatre which holds concerts , music shows and drama presentations. You can also relax by the numerous ponds and fountains at the park or visit the Wole Soyinka Art Gallery to view unique art presentations. The Lagos Marina is host to a number of office buildings, and other structures such as the Bookshop House which was formerly owned by CMS and the Cathedral Church of Christ . Due to

2850-494: The region as a result of blocked drainage. As a result of the poor waste management in the area, the environment in the iconic square looks quite unkempt. See also: List of schools in Lagos See also: List of hospitals in Lagos See also: List of festivals in Lagos See also: Architecture of Lagos 6°26′43″N 3°24′07″E  /  6.44530°N 3.40194°E  / 6.44530; 3.40194 Lagos Island Lagos Island ( Yoruba : Ìsàlẹ̀ Èkó )

2907-425: The rule of unique invertibility") determined the structure of finite simple semigroups and showed that the minimal ideal (or Green's relations J-class) of a finite semigroup is simple. From that point on, the foundations of semigroup theory were further laid by David Rees , James Alexander Green , Evgenii Sergeevich Lyapin  [ fr ] , Alfred H. Clifford and Gordon Preston . The latter two published

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2964-406: The same structure. A semigroup congruence ~ is an equivalence relation that is compatible with the semigroup operation. That is, a subset ~ ⊆ S × S that is an equivalence relation and x ~ y and u ~ v implies xu ~ yv for every x , y , u , v in S . Like any equivalence relation, a semigroup congruence ~ induces congruence classes and the semigroup operation induces

3021-414: The semigroup operation is commutative, then the semigroup is called a commutative semigroup or (less often than in the analogous case of groups ) it may be called an abelian semigroup . A monoid is an algebraic structure intermediate between semigroups and groups, and is a semigroup having an identity element , thus obeying all but one of the axioms of a group: existence of inverses is not required of

3078-436: The term maximal subgroup differs from its standard use in group theory. More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at least one idempotent. The number of finite semigroups of a given size (greater than 1) is (obviously) larger than the number of groups of the same size. For example, of the sixteen possible "multiplication tables" for

3135-431: The unique one-sided identity). A semigroup S without identity may be embedded in a monoid formed by adjoining an element e ∉ S to S and defining e ⋅ s = s ⋅ e = s for all s ∈ S ∪ { e } . The notation S denotes a monoid obtained from S by adjoining an identity if necessary ( S = S for a monoid). Similarly, every magma has at most one absorbing element , which in semigroup theory

3192-670: Was home to the Brazilian Quarter of Lagos where the majority of the slave trade returnees from Brazil stayed. Many families lived on Broad Street in the Marina. The poor eastern side of the island contains the main markets and poor housing and is overcrowded. It is the part of Lagos where the Oba (king) of Lagos resides. It is also believed that the Eyo festival can only be held in this part of Lagos. Most Nigerian banks' head offices are located on Lagos Island. First Bank of Nigeria

3249-468: Was later demolished by the government of Yakubu Gowon to make way for Tafawa Balewa Square. In its hey days, the course hosted the Empire Day parades. The horse racing track was about seven to eight furlongs or a mile. In 1960, the course was redeveloped to celebrate Nigeria's independence and the lowering of the union jack. TBS was constructed in 1972 over the site of a defunct rack for horse racing. It

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