*(English)*Zbl 0855.20054

Continuing earlier papers the author again considers Morita equivalent semigroups, i.e. semigroups $R$ and $S$ such that the categories $R$-**FxAct** and $S$-**FxAct** are equivalent. Here $S$-**FxAct** consists of unitary left $S$-acts ${}_{S}M$ such that the canonical $S$-homomorphism ${{\Gamma}}_{M}:S\otimes S({}_{S}S,{}_{S}M)$ defined by $s\otimes t\alpha \mapsto \left(st\right)\alpha $ is an isomorphism and ${}_{S}M$ is called unitary if $SM=M$. A semigroup $S$ which is unitary as a left $S$-act, i.e. fulfills $S={S}^{2}$ is called factorizable.

A six-tuple $\langle R,S,{}_{R}{P}_{S},{}_{S}{Q}_{R},\langle ,\rangle ,[,]\rangle $ is said to be a unitary Morita context, where $R$ and $S$ are factorisable semigroups, ${}_{R}{P}_{S}$ and ${}_{S}{Q}_{R}$ are unitary $R$-$S$- and $S$-$R$-biacts respectively, $\langle ,\rangle $ is an $R$-$R$-morphism of $P\otimes {}_{S}Q$ into $R$, and $[,]$ is an $S$-$S$-morphism of $Q\otimes {}_{R}P$ into $S$ such that the following hold: 1. $\langle p,q\rangle {p}^{\text{'}}=p[q,{p}^{\text{'}}]$ 2. $q\langle p,{q}^{\text{'}}\rangle =[q,p]{q}^{\text{'}}$.

The author calls two semigroups $R$ and $S$ strongly Morita equivalent if there exists a unitary Morita context such that $\langle ,\rangle $ and $[,]$ are surjective. It is proved that strong Morita equivalence implies Morita equivalence. A semigroup is called a sandwich semigroup if there exists a set of idempotents $E$ in $S$ such that $S=SES$. If $R$ and $S$ are sandwich semigroups which are strongly Mority equivalent, then the subcategories of principal projectives are equivalent and the cardinalities of sets of regular $D$-classes of $R$ and $S$ are equivalent. If $e,f\in R$ are idempotent, then there exist idempotents ${e}^{\text{'}},{f}^{\text{'}}\in S$ such that $eRf$ and ${e}^{\text{'}}S{f}^{\text{'}}$ are isomorphic semigroups.

Let $R$ be a semigroup and ${}_{R}P$, ${Q}_{R}$ be left and right $R$-acts respectively and $\langle ,\rangle :{}_{R}P\times {Q}_{R}\to R$ a mapping such that $\langle rp,q\rangle =r\langle p,q\rangle $ and $\langle p,qr\rangle =\langle p,q\rangle r$ for $p\in P$, $q\in Q$, $r\in R$. Then the tensor product ${Q}_{R}\otimes {}_{R}P$ becomes a semigroup with respect to the product $(q\otimes p)({q}^{\text{'}}\otimes {p}^{\text{'}})=q\otimes \langle p,{q}^{\text{'}}\rangle {p}^{\text{'}}$. This semigroup is called the Morita semigroup over $R$ defined by $\langle ,\rangle $.

It is proved that ${Q}_{R}\otimes {}_{R}P$ is strongly Morita equivalent to $R$ if $R$ is factorizable, $P$ and $Q$ are unitary and $\langle ,\rangle $ is surjective. As an application the author shows that a completely 0-simple semigroup gives rise to a Morita context. Then using the result of a former paper of the author (cited as [14]) that a completely 0-simple semigroup is Morita equivalent to a group with 0, he gets a new proof of the classical Rees structure theorem.

##### MSC:

20M50 | Connections of semigroups with homological algebra and category theory |

20M10 | General structure theory of semigroups |