morphism


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morphism

[′mȯr‚fiz·əm]
(mathematics)
The class of elements which together with objects form a category; in most cases, morphisms are functions which preserve some structure on a set.
References in periodicals archive ?
By an endomorphism f: X [right arrow] X, we mean a morphism from a projective variety X to itself.
The coproduct of (A, f, B),(C, g, B) [member of] Ob(S(B)) is a triple ((A [??] C, h, B), [alpha], [beta]), where (A [??] C, h, B) [member of] Ob(S(B)), [alpha] [member of] Mor((A, f, B),(A [??] C, h, B)), [beta] G Mor((C, g, B),(A [??] C, h, B)) such that for every (X, j, B) [member of] Ob(S(B)) and every pair of morphisms [gamma] [member of] Mor((A, f, B),(X, j, B)) and [delta] [member of] Mor((C, g, B),(X, j, B)) there exists a unique morphism [theta] [member of] Mor((A [??] C, h, B),(X, j, B)) such that [theta] [??] [alpha] = [gamma] and [theta] [??] [beta] = [delta]
Let (A, *) be a Maltsev algebra and [alpha] : A [right arrow] A an algebra morphism. Let [[alpha].sup.0] = id and, for any integer n [greater than or equal to] 1, [[alpha].sup.n] = [alpha] [??] [[alpha].sup.n-1].
A morphism of left (A, [alpha])-Hom-modules is a morphism of left A-modules in the Hom-category [??]([M.sub.k]).
Suppose that Z is another compact ENR and that b/q is a morphism from Y to Z prescribed by a fibrewise manifold q : [~.Y] [right arrow] Y with fibres of dimension n and a map b : [~.Y] [right arrow] Z.
Consider the identity morphism [G.sub.1] to [G.sub.1].
Let [phi] : ([X.sub.1], [f.sub.1]) [right arrow] ([X.sub.2], [f.sub.2]) be a morphism of dynamical systems.
A morphism of a direct system [{[P.sub.i], [r.sup.i.sub.j]}.sub.I] to a direct system [{[Q.sub.i'], [[rho].sup.i'.sub.j']}.sub.I'] consists of an order preserving map f: I [right arrow] I' and A-module morphisms [[PHI].sub.i]: [P.sub.i] [right arrow] [Q.sub.f(i)] which obey compatibility conditions [[rho].sup.f(i).sub.f(j)] [omicron] [[PHI].sub.i] = [[PHI].sub.J] [omicron] [r.sup.i.sub.j].
Then f: (X, [A.sub.X]) (Y, [A.sub.Y]) is called a morphism if [A.sub.X] [subset] [f.sup.-1] ([A.sub.Y]), i.e.,
We illustrate the previous notions on the Thue--Morse word t, the fixed point of the morphism 0 [??] 01 and 1 [??] 10 starting with 0, i.e., t = 011010011001011010 x x x.
Indeed, by the cyclicity (7.2.3) of the tensor factor [??] of [mathematical expression not reproducible], and by the structure theorem (6.3) of the tensor factor [??] of [mathematical expression not reproducible], the morphism [E.sub.p] of (7.2.6) is well-defined from [mathematical expression not reproducible] "onto" [??], for p [member of] P.
Histological and immunohistochemical examinations revealed that the tumor had inconspicuous nuclear morphism and plenty of renin granules in the cytoplasm.