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Residuated mapping
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<p>In mathematics, the concept of a residuated mapping arises in the theory of <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered sets</a>. It refines the concept of a <a href="/facts/Monotonic_function/MjQK0YL2">monotone function</a>. </p><p>If <i>A</i>, <i>B</i> are <a href="/facts/Partially_ordered_sets/qb4a3FAX">posets</a>, a function <i>f</i>: <i>A</i> → <i>B</i> is defined to be monotone if it is order-preserving: that is, if <i>x</i> ≤ <i>y</i> implies <i>f</i>(<i>x</i>) ≤ <i>f</i>(<i>y</i>). This is equivalent to the condition that the <a href="/facts/Preimage/KANefMAl">preimage</a> under <i>f</i> of every <a href="/facts/Down-set/hSyGJ2WB">down-set</a> of <i>B</i> is a down-set of <i>A</i>. We define a <a href="/facts/Principal_down-set/hSyGJ2WB">principal down-set</a> to be one of the form ↓{<i>b</i>} = { <i>b</i>' ∈ <i>B</i> : <i>b</i>' ≤ <i>b</i> }. In general the preimage under <i>f</i> of a principal down-set need not be a principal down-set. If all of them are, <i>f</i> is called residuated. </p><p>The notion of residuated map can be generalized to a <a href="/facts/Binary_operator/CYtow0bz">binary operator</a> (or any higher <a href="/facts/Arity/zSVuwHvP">arity</a>) via component-wise residuation. This approach gives rise to notions of left and right division in a partially ordered <a href="/facts/Magma_(algebra)/9g3vp7Wl">magma</a>, additionally endowing it with a <a href="/facts/Quasigroup/LRWCA2oB">quasigroup</a> structure. (One speaks only of residuated algebra for higher arities). A binary (or higher arity) residuated map is usually <i>not</i> residuated as a unary map. </p>