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In mathematics, the four-spiral semigroup is a special semigroup generated by four idempotent elements. This special semigroup was first studied by Karl Byleen in a doctoral dissertation submitted to the University of Nebraska in 1977. It has several interesting properties: it is one of the most important examples of bi-simple but not completely-simple semigroups; it is also an important example of a fundamental regular semigroup;〔 it is an indispensable building block of bisimple, idempotent-generated regular semigroups.〔 A certain semigroup, called double four-spiral semigroup, generated by five idempotent elements has also been studied along with the four-spiral semigroup.〔〔 ==Definition== The four-spiral semigroup, denoted by ''Sp4'', is the free semigroup generated by four elements ''a'', ''b'', ''c'', and ''d'' satisfying the following eleven conditions:〔 : * ''a''2 = ''a'', ''b''2 = ''b'', ''c''2 = ''c'', ''d''2 = ''d''. : * ''ab'' = ''b'', ''ba'' = ''a'', ''bc'' = ''b'', ''cb'' = ''c'', ''cd'' = ''d'', ''dc'' = ''c''. : * ''da'' = ''d''. The first set of conditions imply that the elements ''a'', ''b'', ''c'', ''d'' are idempotents. The second set of conditions imply that ''a R b L c R d'' where ''R'' and ''L'' are the Green's relations in a semigroup. The lone condition in the third set can be written as ''d'' ωl ''a'', where ωl is a biorder relation defined by Nambooripad. The diagram below summarises the various relations among ''a'', ''b'', ''c'', ''d'': 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「Four-spiral semigroup」の詳細全文を読む スポンサード リンク
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