On the semigroup of fully indecomposable relations
Chong-Yun Chao, Mou‐Cheng Zhang · Czechoslovak Mathematical Journal · 1983
The purpose of this note is to give a sufficient condition for the conjecture in [4] concerning the semigroup of fully indecomposable relations to hold.A binary relation on a finite set 0" = {a^, «2,..., a"] of n elements, n > l,is a subset of 0" x Q" = {(a,-, aj); a^, aj e Q"}, Let В = B(Q^) be the set of all (binary) relations on Q".Then Б is a semigroup with the multiplication defined as follows: for Q and T in B, (a^, aj) e ^т if there is я aj, G Q" such that (a^-, a^) e Q and (a^, Uj) e т.Let (o be the universal relation on Q", i.e., со = Q" x Q".Let M" denote the set of all w X и matiices over the Boolean algebra of {0, l}.Then M" is a semigroup under the ordinary matrix multiplication, and the map