TURNING RETRACTIONS OF AN ALGEBRA INTO AN ALGEBRA
Dragan Mašulović · 2004
One can turn the set of retractions of a lattice hL;•i into a poset Rf(L) by letting fg i f(x) • g(x) for all x 2 L. In 1982 H. Crapo raised the following two problems: (1) Is it true that Rf(L) is a lattice for any lattice L? (2) Is it true that Rf(L) is a complete lattice if L is a complete lattice? In 1990 and 1991 B. Li published two papers dealing with the above two questions. He showed that Rf(L) is not necessarily a lattice and that L is a complete lattice if and only if Rf(L) is a complete lattice. Motivated by the idea of extending the structure from the base set to the set of all retractions, we introduce the notion of R-algebra as follows. Let Rf(A) denote the set of all retractions of an algebra A. We say that A is an R-algebra if the set Rf(A) is closed with respect to operations of A applied pointwise. We give some necessary and some sucient conditions for A to be an R-algebra. We show that the property of being an R- algebra carries over to retracts of the algebra. In a set of examples we show that almost no classical algebra is an R-algebra. In particular, a lattice L is an R-algebra i jLj • 2.