Some algebraic properties of b-parts of real numbers


Mohamad Hadi HOOSHMAND, Hailiza Kamarul HAILI

AMS Classification:

11A99, 11A67


additive real group, b-addition, b-additive real set, b-bounded group, b-decimal and b-integer part of real numbers, b-symmetric real set, b-subtractive real set, decimal group, generalized division algorithm


The b-parts of real numbers were introduced and studied in [2] and [3] by M.H. Hooshmand. In this paper, we first discuss b-parts of real numbers and give another proof for the generalized division algorithm introduced in [3]. Then we focus on b-addition of two real numbers as a new binary operation in \mathbb{R}. In fact, b-addition of real numbers is b-decimal part of their ordinary addition denoted by +_b. We show that (\mathbb{R},+_b) is a semigroup and determine its subgroups (subsets which are groups with +_b). We call them b-bounded groups, because they are subsets of \mathbb{R}_b=b[0,1). Finally, we discuss the relations between b-bounded groups and subgroups of the additive real group and its quotient groups.

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Vol. 3 (11), 2008