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residuated mapping
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 it is order-preserving: that is, if ''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 ↓ = . In general the preimage under ''f'' of a principal down-set need not be a principal down-set. If it is, ''f'' is called residuated.
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.〔Denecke, p. 95; Galatos, p. 148〕
==Definition==
If ''A'', ''B'' are posets, a function ''f'': ''A'' → ''B'' is residuated if and only if the preimage under ''f'' of every principal down-set of ''B'' is a principal down-set of ''A''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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