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In mathematics, a refinement monoid is a commutative monoid ''M'' such that for any elements ''a0'', ''a1'', ''b0'', ''b1'' of ''M'' such that ''a0+a1=b0+b1'', there are elements ''c00'', ''c01'', ''c10'', ''c11'' of ''M'' such that ''a0=c00+c01'', ''a1=c10+c11'', ''b0=c00+c10'', and ''b1=c01+c11''. A commutative monoid ''M'' is ''conical'', if ''x''+''y''=0 implies that ''x''=''y''=0, for any elements ''x'',''y'' of ''M''. == Basic examples == A join-semilattice with zero is a refinement monoid if and only if it is distributive. Any abelian group is a refinement monoid. The positive cone ''G+'' of a partially ordered abelian group ''G'' is a refinement monoid if and only if ''G'' is an ''interpolation group'', the latter meaning that for any elements ''a0'', ''a1'', ''b0'', ''b1'' of ''G'' such that ''ai ≤ bj'' for all ''i, j<2'', there exists an element ''x'' of ''G'' such that ''ai ≤ x ≤ bj'' for all ''i, j<2''. This holds, for example, in case ''G'' is lattice-ordered. The ''isomorphism type'' of a Boolean algebra ''B'' is the class of all Boolean algebras isomorphic to ''B''. (If we want this to be a set, restrict to Boolean algebras of set-theoretical rank below the one of ''B''.) The class of isomorphism types of Boolean algebras, endowed with the addition defined by (for any Boolean algebras ''X'' and ''Y'', where denotes the isomorphism type of ''X''), is a conical refinement monoid. 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「refinement monoid」の詳細全文を読む スポンサード リンク
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