Partial Multipliers on Partially Ordered Sets
نویسنده
چکیده
According to Larsen [15, p. 13], a function F from a nonvoid subset DF of a commutative semigroup A into A is called a partial multiplier on A if F (D) · E = F (E) ·D for all D, E ∈ DF . Note that if in particular DF is a subsemigroup of A and F ( D · E ) = F (D) · E for all D , E ∈ DF , then F is a partial multiplier on A . A partial multiplier F on A is called total if DF = A . Clearly, the identity function ∆A of A is a total multiplier on A . Moreover, if A ∈ A , then the function FA, defined by FA(D) = A · D for all D ∈ A , is also a total multiplier on A . A total multiplier F on A is called inner if F = FA for some A ∈ A . Note that each total multiplier on A is inner if and only if A has an identity element. The investigation of total multipliers is completely justified by Larsen [15]. To motivate the appropriateness of considering partial multipliers on A , we make here a slight modification of the classical definition of fractions in accordance with the ideas of Lambek [14, p. 36] .
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