Math, Ann. 184, 78 (1969)

N o t e o n a T h e o r e m o f Pultr CRAIG R . PLATT

The title refers to [ i ] , in which certain full embeddings of categories of algebras are described. Here a simpler form of the strong embedding of 9.I(1, 1) into 91(1, 1, 0) is given (Lemma 7 of [1]). We assume familiarity with the concepts and notations of [ 1]. Let F = V2° K2. Thus, for a set X, F(X)= (X × {0, 1 }) v {0, 1}. For A = (X, q~, ~,) in 9.I(1, 1) define ~(A) = (F(X), c~,fl, c) in 2[(1, i, 0) as follows: c~(0)=fl(0)=l; a ( t ) = f l ( 1 ) = 0 ; c is 0; if x e X , then e((x, 0)) = (x, 1), fl((x, 0)) = (x, 0), a((x, 1)) = (~o(x), 0), and fl((x, 1)) = (tp(x), 0). Then q~defines a unique 1 - 1 functor such that [] o q~ = F o fq. Let Ai = (X~, ~0i,~o~) be objects in 9t(1, 1), and ~(Ai) = (F(Xi), el, fit, ci) for i e {0, 1}. Let 9 : ~(Ao) 4~(A1) be a morphism in 9/(1, 1, 0). Then O(0) = g(Co) = c~ = 0, so 9(1) = g(ao(0)) = a I (g(0)) = cq (0) = 1. I f x s Xo, then g((x, 0)) = 9(fio((X, 0))) = [h (9(( x, 0))), hence 9((x, 0)) e X 1 x {0}, since these are the only elements fixed under ill. Thus, we can define f:Xo--*X1 uniquely by requiring g ( ( x , 0 ) ) = (f(x), 0). Then g((x, 1)) = g(~o((X, 0))) = ~1(9((x, 0))) = oq((f(x), 0)) = (f(x), 1). Hence, (f(~Oo(X)), 0 ) = O(~po(x), 0 ) = g(~o((x, 1 ))) = ~1 (g((x, 1))) = oq((f(x), 1 )) = ( q h ( f ( x ) ) , 0 ) . Thus, f o ~ o o = q h o f . Similarly f o t p o = q q o f , so f is a morphism in 91(1, 1). Since F(f) = 9 is clear, q, is a full embedding.

References 1. Pultr, A. : Eine Bemerkung fiber voile Einbettungen yon Kategorien von Algebren~ Math. Ann. 178, 78--82 (1968), Professor Craig Platt Department of Mathematics University of Manitoba Winnipeg, Canada

(Received March 4, 1969)