Paper Information

Title: 

F-MULTIPLICATION OF BINARY OPERATIONS AND DECOMPOSITIONAL GROUPS

Type: PAPER
Author(s): HOUSHMAND M.H.*,HAILI H.K.
 
 *SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY SCIENCE OF MALAYSIA 11800 MINDEN, PENANG, MALAYSIA
 
Name of Seminar: SEMINAR ON ALGEBRA (SALG20)
Type of Seminar:  CONFERENCE
Sponsor:  TARBIAT MOALLEM UNIVERSITY, FACULTY OF MATHEMATICAL SCIENCES AND COMPUTER
Date:  2009Volume 20
 
 
Abstract: 

The b-parts of real numbers were introduced and studied in [4], and b-addition of real numbers in [3]. The b-parts have many interesting number theoretic explanations, analytic and algebraic properties and the b-addition is b-decimal part of ordinary addition of two real numbers and is denoted +b. It was shown that (R,+b) is a semigroup (equivalently the b-decimal part function ( )b satisfies the associative functional equation: f(x+f(y+z)) = f(f(x+y)+z)) and Rb = b[0, 1) is its largest subgroup. As a generalization of this topic, decomposer and associative functions on binary systems (magmas), semigroups and groups are introduced and studied in [2]. If (X, ·) is a binary system and f an arbitrary function from X to X, then another binary operation (in X) is defined by x ·f y = f(xy). The binary system (X, ·f) is a semigroup if and only if f is associative in (X, ·). In [2] we solved associative equations in arbitrary groups and proved that the associative equation does not have any nontrivial solutions in the simple groups. In this way all associative binary operations ·f for a group (G, ·) are characterized.
In this talk I discuss about the f-multiplication and the semigroup (G, ·f), where f: G
®G is associative, and show that (f(G), ·f) is a group, namely f-decompositional group, and it is the largest subgroup of the semigroup (G, ·f ). Also some other properties of f-decompositional groups will be considered.

 
Keyword(s): BINARY OPERATION, ASSOCIATIVE FUNCTION, F-MULTIPLICATION, DECOMPOSITIONAL GROUP, B-INTEGER AND B-DECIMAL PARTS OF REAL NUMBERS, B-PARTS REAL FUNCTIONS, B-ADDITION, B-BOUNDED GROUP, DECIMAL GROUP, FUNCTIONAL EQUATION
 
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