Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context （S,R,<sub>S</sub>P<sub>R</sub>,<sub>R</sub>Q<sub>S</s...Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context （S,R,<sub>S</sub>P<sub>R</sub>,<sub>R</sub>Q<sub>S</sub>,） with and surjective.For a factorisable semigroup S,we denote ζ<sub>S</sub>={（s<sub>1</sub>,s<sub>2</sub>）∈S×S|ss<sub>1</sub>=ss<sub>2</sub>,<sub>S</sub>∈S},S′=S/ζ<sub>S</sub> and US-FAct={<sub>S</sub>M∈ S-Act|SM=M and SHom<sub>S</sub>（S,M）≌M}.We show that,for factorisable semigroups S and R,the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S′ and R′ are strongly Morita equivalent.Some conditions for a factorisable semigroup to be strongly Morita equivalent to a sandwich semigroup,local units semigroup,monoid and group separately are also given.Moreover,we show that a seinigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I,SSHom<sub>S</sub>(S,<sub>i∈I</sub>S)→<sub>i∈I</sub>S,st·f（st）f is an S-isomorphism.展开更多
基金The research is partially supported by a UGC(HK) grant ≠2160092Project is supported by the National Natural Science Foundation of China
文摘Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context （S,R,<sub>S</sub>P<sub>R</sub>,<sub>R</sub>Q<sub>S</sub>,） with and surjective.For a factorisable semigroup S,we denote ζ<sub>S</sub>={（s<sub>1</sub>,s<sub>2</sub>）∈S×S|ss<sub>1</sub>=ss<sub>2</sub>,<sub>S</sub>∈S},S′=S/ζ<sub>S</sub> and US-FAct={<sub>S</sub>M∈ S-Act|SM=M and SHom<sub>S</sub>（S,M）≌M}.We show that,for factorisable semigroups S and R,the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S′ and R′ are strongly Morita equivalent.Some conditions for a factorisable semigroup to be strongly Morita equivalent to a sandwich semigroup,local units semigroup,monoid and group separately are also given.Moreover,we show that a seinigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I,SSHom<sub>S</sub>(S,<sub>i∈I</sub>S)→<sub>i∈I</sub>S,st·f（st）f is an S-isomorphism.
