Decomposer Type Functions on Groups
M. H. Hooshmand *
Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran.
*Author to whom correspondence should be addressed.
Abstract
Decomposer functions in algebraic structures were introduced and studied in the paper ”Decomposer and associative functional equations” in 2007. If (G, ) is a group, and f : G → G is a map, then f is called a right [resp. left] decomposer if and only if f(f*(x)f(y)) = f(y) [resp. f(f(x)f*(y)) = f(x)] for all x, y ∈ G, where f*(x)f(x) = f(x)f*(x) = x. Also, f is called a decomposer if it is left and right decomposer. There are many important connections between these functions and decomposition of groups by subsets. Now, we observe that if the structure is a group, then there are more important properties for them and also many connections among decomposer functions, multiplicative symmetric functions (introduced by J. G. Dhombres in 1973), separator functions and so on. For instance, every idempotent endomorphism in groups is (strong) decomposer. We also introduce some other related types and generalizations of these functions such as semi-strong decomposer and weak decomposer which help us in the study of decomposer type functions in groups. Then, several important properties and relations for these functions will be proved. Finally, we completely characterize the (two-sided) decomposer functions in arbitrary groups, and so we give a general solution of the decomposer equations that were not solved in the paper 2007 (only left and right cases were characterized).
Keywords: Decomposer function, multiplicative symmetric function, subset projection, functional equation on group