DIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS

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1 GANIT J. Bangladesh Math. oc. IN ) 0) -7 DIRECT PRODUCT AND WREATH PRODUCT OF TRANFORMATION EMIGROUP ubrata Majumdar, * Kalyan Kumar Dey and Mohd. Altab Hossain Department of Mathematics University of Rajshahi, Rajshahi-605, Bangladesh * kkdmath@yahoo.com Received Accepted ABTRACT In this paper direct product and wreath product of transformation semigroups have been defined, and associativity of both the products and distributivity of wreath product over direct product have been established. Keywords: Transformation semigroup, Direct product, Wreath product. Introduction Direct product and wreath product of transformation groups are well known see [,5]). We have generalized these products to transformation semigroups. We have proved that both direct product and wreath product are associative, and that wreath product is distributive over direct product.. Direct Product and Wreath Product Definition. Let be a semigroup and X a non-empty set. will be called a transformation semigroup on X if there is a mapping φ: X X, for which we write φs, ) = s) and which satisfies the condition s s )) =s s )), for each X and for each s, s. If is a monoid, i.e., if has an identify element then the mapping φ is further assumed to satisfy ) =, for each X. For every transformation semigroup on X, there is a homomorphism ψ : EX), the semigroup of all mappings f : X X, given by ψs) = f, where f) = s). EX) is usually called the full transformation semigroup on X. Let X and X be two non-empty disjoint sets and let and be transformation semigroups on X and X respectively.

2 Majumdar et al. Definition. The direct product of and, written, is defined as a transformation semigroup on X X, the elements of being the ordered pairs s, s ), s, s, with s, s ) ) = s ),s, s ) ) = s ), for each X, X. The multiplication in is component-wise. It is easily seen that is indeed a transformation semigroup. If and are finite, the number of elements of is obviously the product of the numbers of elements of and. Theorem. If,, are transformation semigroups on X, X, X, then ) ) is a transformation semigroup on X X X. Obviously, the map s, s ), s ) s, s, s )) is an isomorphism of semigroups ) and ). To see that it is also so as transformation semigroups, we note as a typical case, s, s ), s ) ) = s, s ) ) =s ), and also, s, s, s )) ) = s ). Definition. The wreath product of with, written ς is the transformation semigroup on X X consisting of elements θ on X X which are given by θ : X X X X such that θ, ) = s ),s )), with s in and each s in s, s being an element of determined by. It follows from the definition that if,, X, X are finite, then ς = X, where i and X denote the numbers of elements of i i=) and X respectively.. Wreath Product as a Direct Product An equivalent description of wreath product in terms of direct products is given below: Theorem. If, X ) and, X ) are transformation semigroups, then ς, X X ), U X, where each X X, X X and X =. X, Let θ ς, X X ). Then θ, ) = σ ), σ )), for some σ and σ, where σ is in and depends on.

3 Direct Product and Wreath Product Define Φ : ς, X X ), U X U X X X by Φθ)) ) = σ ), Φθ)) ) =σ ). Net, let θ X, U X U X X. Then σ θ ) = σ ) = σ )., for some,, σ depending on and σ. Define Ψ:, U X U X ς, X X ) by X X Ψ θ)), ) = σ ), σ )). If θ ς, X X ) is given by θ, ) = σ ), σ )), where σ and σ and depends on, then ϕ θ ) ) = σ ) ϕ θ ) ) = σ ) and θ θ ) ) = σ σ ) ), σ σ ) )). Hence ϕ θθ ) ) = σ σ ) ) ϕ θθ ) ) = σ σ ) )). Also, ϕ θ ) ϕ θ )) ) = σ σ ) ) ϕ θ ) ϕ θ )) ) = σ σ ) ) Φθθ ) = ΦΨθ)Φθ ). Thus ϕ is a homomorphism. If θ, U X U X X X θ ) = σ ) = σ ) then ϕ θ ), ) = σ ), σ )). Also, θ θ ) ) = σ σ ) ) θ θ ) ) = σ σ ) ) is given by

4 4 Majumdar et al. Hence ψ θθ )), ) = σ σ ), σ σ )) and ψ θ ) ψ θ )) ) = σ σ ), σ σ )) so that Ψθθ ) = Ψθ)Ψθ ) i.e., Ψ is a homomorphism. Now ϕψ ) θ ) and ϕψ ) θ ) ΦΨ)θ) = θ. ϕψ = ) = ϕ ψ θ )) ) = ϕ ψ θ )) ) = σ. X ) = σ ), )), U X U X. X Also, ΨΦ)θ), ) = ΨΦθ)), ) = σ ), σ )) = θ, ). ΨΦθ) = θ, and so ΨΦ =. ς, X X ) Thus Φ and Ψ are inverses of each other. Therefore, both Φ and Ψ are isomorphisms. The following remarks are very significant and useful. Remarks i) If, X ) and, X ) are transformation semigroups with = { },, then, X X ) and ς, X X ) may be identified with, X ) and,, U X respectively, ignoring the trivial action of on X. Here, each X X, and each X is in - correspondence with X with and, s ) s ). Thus, in this case, = and ς direct X product) as semigroups. ii) If ={ }, then both, X X ) and ς, X X ) may be identified with, X ) since = s =, for each pair of elements, X. s, iii) If and are transformation semigroups on the same set X, then ς, X X) may be identified with, UX X. As semigroups, ς. X X X If, in particular, = and X is finite with X = n, then ς L L n + copies).

5 Direct Product and Wreath Product 5 4. Associativity of Wreath Products Theorem 4. If, X ),, X ),, X ) are three transformation semigroups, then ς ) ς, X X ) X ) ς ς ), X X X ). Define ϕ : ς ) ς ς ς ) and ψ : ς ς ) ς ) ς as follows: If θ ς ) ς ) is given by θ, ), ) = α σ ) ) where α ) σ ),, σ,, ς, depends on and is defined by α = )) so that θ, ), )= σ, ), σ, )), σ )), then ϕθ) is given by, ϕθ),, ))= σ,, ), σ, )), σ )). Also, if θ ς ς ) is given by θ,, )) = σ, ) ), σ, )) = σ, ), σ, ), σ ))),, then ψθ) = σ,, ), σ, ), σ ))., If θ and θ are defined similarly with the σ's and σ's replacing by σ's and σ's then θθ and θ θ are given by θθ ), ), ) =θ σ, ), σ, )), σ )), = σ, σ ) ), σ, σ, ) )), σ σ ) )), and θ θ ), ), )) = θ σ, ), σ ), σ )), = σ σ, ), σ σ, ), σ σ )), It is clear that ϕθθ ) = ϕθ)ϕθ ) and ψθ θ ) = ψθ)ψθ ), i.e, ϕ and ψ are homomorphisms. Also it is evident from the definitions of ϕ and ψ that they are inverses of each other. Hence both ϕ and ψ are isomorphisms. 5. Distributivity of Wreath Products over Direct Products The following isomorphism theorem may be viewed as showing that wreath product of the stated type is distributive over as a direct product that arises in a natural manner.

6 6 Majumdar et al. Theorem 5. Let, X ),, X ) and, X ) be three transformation semigroups. Then ς ), X X X )) ς ) ς ), X X ) X X )). Define ϕ : ς ), X X X )) ς ) ς ), X X ) X X )) by ϕ θ) = θ, θ ) ) where, if θ, ) = σ ), σ, σ ) )) = σ ), σ )), ) and θ, ) = σ ), σ, σ ) )) = σ ), σ )), ) then θ,θ ), ) = θ, ) = σ ), σ )) 4) θ,θ ), ) = θ, ) = σ ), σ )) 5) Also define ψ: ς ) ς ), X X ) X X )) ς ), X X X )) by ψ θ,θ )=θ where if θ, θ ), ) = θ, ) = σ ), σ )), θ, θ ), ) = θ, ) = σ ), σ )), 6) then θ, ) = σ ), σ, σ ) ) = σ ), σ )) 7) θ, ) = σ ), σ, σ ) ) = σ ), σ )). 8) It follows from ) - 8) that ϕψ = ς ) ς ), X X ) X X )) X and ψϕ = ς ), X X )) Thus both ϕ and ψ are - and onto. Now, let θ, θ ς ), X X X )) by given by θ, ) = σ ), σ )) θ, ) = σ ), σ )) where σ, σ, and σ, σ the former being determined by and the latter by and θ, ) = σ ), σ )), θ, ) = σ ), σ ))

7 Direct Product and Wreath Product 7 where σ, σ, and σ, σ latter by. Then θ θ), ) = σ σ ), σ σ )), ) θ θ), ) = σ σ, σ σ ))., ince, ϕ θ) = θ,θ )andϕ θ) = θ θ ), where θ θ ), ) = σ ), σ )), θ θ ), ) = σ ), σ )), and θ θ ), ) = σ ), σ )), θ θ ), ) = σ ), σ ))., We have ϕ θ θ) = ϕ θ) ϕ θ). the former being determined by and the Hence ϕ is a homomorphism. Therefore ϕ is an isomorphism. Thus ς ), X X X )) ς ) ς ), X X ) X X ). Application of direct product and wreath product of transformation groups and transformation semigroups appear in [5,6]. REFERENCE. Hall, M., Group Theory, Macmillan Co. New York, Karpuz, E. G.and Cevik, A.., The word and generalized word problem for semigroups under wreath products, Bull. Math. oc. ci. Math. Roumanie Tome 5, 00, 009) Maclane,. and Birkhoff, G., Algebra. Macmillan Co. New York, Majumdar,., The auotomorphism group of a mapping, M.c. Thesis, Univ. of Birmingham, Meldrum, J. D. P., Wreath products of groups and semigroups, Longman, Robertson, E. F., Ruskuc, N. and Thomson, M. R., On finite generation and other finiteness conditions for wreath products of semigroups, Comm. Algebra, 0 00)