On coproducts of rectangular bands
H. E. Scheiblich · Rocky Mountain Journal of Mathematics · 1978
For rectangular bands X and Y, a characterization is given of the coproduct of X and Y in the category of rectangular bands and in the category of all bands.A characterization is also given of the coproduct of X and Y amalgamating a subband U. Introduction.A semigroup is a band provided x 2 = x for all x, and a band is rectangular provided xz = xyz for all x, y, and z.A band X is rectangular if and only if X is the direct product of a left zero semigroup L and a right zero semigroup R[2, Theorem 4.1.5],The main structure theorem for bands is that every band is a semilattice of rectangular bands [1].More specifically, for a band B, Green's relation Ï& is a congruence on B, B/£> is a semilattice, and each Sodass is a rectangular band [2, Corollary 7.4.7].When X i9 X 2 are bands, the coproduct (free product) of X x and X 2 in the category !B of all bands is defined to be a band W and a pair of homomorphisms i t : X t ~* W(t = 1, 2) with the mapping property that for each band B and each pair of homomorphisms T t : X t -» B(t = 1, 2), there exists a unique homomorphism 0 : W-> B such that 0° if = T t (t = 1, 2).When X x and X 2 are rectangular bands, the coproduct in the category of rectangular bands (rectangular band coproduct) would be defined by simply replacing band with rectangular band in the definition above.This note will give a characterization of the band coproduct and rectangular band coproduct of two rectangular bands X and Y.The coproduct of X and Y amalgamating a subband U will also be characterized.2. The Rectangular Band Coproduct.LEMMA 2.1.Let X = L X R and Y = P X Ç be rectangular bands and let a : X -> Y be a homomorphism.Then a = (a^ a 2 ) fo r some homomorphisms «! : L -• P and a 2 : Y -* Q.PROOF.Let JL i9 l 2 €E L, let r l9 r 2 G. R and suppose that a(£ 1? r x ) = (Pi> 9i)> Q H) = (?2> 92)-Now if £j = l 2 , then (p 2 , q 2 ) =