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In mathematics, Nambooripad order (also called Nambooripad's partial order) is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies. Since the same partial order was also independently discovered by Robert E Hartwig, some authors refer to it as Hartwig–Nambooripad order. "Natural" here means that the order is defined in terms of the operation on the semigroup. In general Nambooripad's order in a regular semigroup is not compatible with multiplication. It is compatible with multiplication only if the semigroup is pseudo-inverse (locally inverse). == Precursors == Nambooripad's partial order is a generalisation of an earlier known partial order on the set of idempotents in any semigroup. The partial order on the set ''E'' of idempotents in a semigroup ''S'' is defined as follows: For any ''e'' and ''f'' in ''E'', ''e'' ≤ ''f'' if and only if ''e'' = ''ef'' = ''fe''. Vagner in 1952 had extended this to inverse semigroups as follows: For any ''a'' and ''b'' in an inverse semigroup ''S'', ''a'' ≤ ''b'' if and only if ''a'' = '' eb'' for some idempotent ''e'' in ''S''. In the symmetric inverse semigroup, this order actually coincides with the inclusion of partial transformations considered as sets. This partial order is compatible with multiplication on both sides, that is, if ''a'' ≤ ''b'' then ''ac'' ≤ ''bc'' and ''ca'' ≤ ''cb'' for all ''c'' in ''S''. Nambooripad extended these definitions to regular semigroups. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Nambooripad order」の詳細全文を読む スポンサード リンク
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