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John M. Howie, Fundamentals of Semigroup Theory (1995), Clarendon Press, Oxford ISBN 0-19-851194-9 M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs , De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7 .
Its unit element is the class of the ordinary 2-sphere. Furthermore, if a denotes the class of the torus, and b denotes the class of the projective plane, then every element c of the monoid has a unique expression in the form c = na + mb where n is a positive integer and m = 0, 1, or 2. We have 3b = a + b.
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element.
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. [6]
A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton argument. A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale.
History monoids were first presented by M.W. Shields. [1] History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids.
Ordinary monoids are precisely the monoid objects in the cartesian monoidal category Set. Further, any (small) strict monoidal category can be seen as a monoid object in the category of categories Cat (equipped with the monoidal structure induced by the cartesian product).
The FBISE was established under the FBISE Act 1975. [2] It is an autonomous body of working under the Ministry of Federal Education and Professional Training. [3] The official website of FBISE was launched on June 7, 2001, and was inaugurated by Mrs. Zobaida Jalal, the Minister for Education [4] The first-ever online result of FBISE was announced on 18 August 2001. [5]