High Quality Content by WIKIPEDIA articles! In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function. If A, B are posets, a function f: A ? B is defined to be monotone if and only if it is order-preserving: that is, x ? y implies f(x) ? f(y). This is equivalent to the condition that the preimage under f of every down-set of B is a down-set of A. We define a principal down-set to be one of the form ?{b} = {...
High Quality Content by WIKIPEDIA articles! In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function. If A, B are posets, a function f: A ? B is defined to be monotone if and only if it is order-preserving: that is, x ? y implies f(x) ? f(y). This is equivalent to the condition that the preimage under f of every down-set of B is a down-set of A. We define a principal down-set to be one of the form ?{b} = { b' ? B : b' ? b }. In general the preimage of a principal down-set need not be a principal down-set. The notion of residuated map can be generalized to a binary operator (or any higher arity) via component-wise residuation. This approach gives rise to notions of left and right division in a partially ordered magma, additionally endowing it with a quasigroup structure. (One speaks only of residuated algebra for higher arities). A binary (or higher arity) residuated map is usually not residuated as a unary map.
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