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Monoides et semi-anneaux complets. (Complete monoids and semirings). (French) Zbl 0636.16019
Let $$(M,+)$$ be a commutative monoid and assume that $$\sum_ Ix_ i\in M$$ is defined for each family $$(x_ i|$$ $$i\in I)$$ of elements $$x_ i\in M$$, where I is any set [any at most countably finite set]. Then $$(M,+,\sum)$$ is called a complete [c-complete] monoid iff $$\sum$$ extends all finite sums already defined in $$(M,+)$$ and (P)$$\sum_ Ix_ i=\sum_ J(\sum_{I_ j}x_ i)$$ holds for each partition $$I=\cup (I_ j|$$ $$j\in J)$$ of I. Correspondingly, let $$(M,+,\cdot)$$ be a semiring which need not have an identity, but an additive neutral 0 which is multiplicatively absorbing. Then $$(M,+,\cdot,\sum)$$ is called a complete [c-complete] semiring iff $$(M,+,\sum)$$ is a complete [c- complete] monoid and z($$\sum_ ix_ i)=\sum_ i(zx_ i)$$ as well as $$(\sum_ Ix_ i)z=\sum_ I(x_ iz)$$ are satisfied for all $$z\in M$$ and all families $$(x_ i|$$ $$i\in I)$$ under consideration. Moreover, a suitable topology $${\mathcal T}$$ on $$(M,+)$$ can be used to define $$\sum_ Ix_ i$$ such that $$(M,+,\sum)$$ is a complete [c-complete] monoid, then called t-complete [tc-complete], and likewise for semirings $$(M,+,\cdot).$$
Various statements concerning these eight concepts and corresponding (counter-)examples are given. For instance, complete [c-complete] does not imply t-complete [tc-complete], each complete monoid has an additively absorbing element, and each tc-complete monoid (semiring) has a natural partial order.
Referee’s remarks: 1) The paper refers to S. Eilenberg [Automata, Languages and Machines (1974; Zbl 0317.94045)] and W. Kuich [Lect. Notes Comput. Sci. 267, 212-225 (1987; Zbl 0625.16026)]. But there and elsewhere (cf. below) complete [c-complete] has another meaning, since instead of the axiom (P) above a stronger one, say (GP), is assumed where partitions (all $$I_ j\neq \emptyset)$$ are replaced by generalized partitions $$(I_ j=\emptyset$$ may occur). In fact, t-complete [tc- complete] monoids and semirings satisfy (GP) as well. 2) There are also papers dealing with complete [c-complete] monoids and semirings as a special case of partial complete monoids and semirings. Corresponding hints and references are given in H. J. Weinert [Lect. Notes Math. 1320, 380-416 (1988)].
Reviewer: H.J.Weinert

##### MSC:
 16Y60 Semirings 20M14 Commutative semigroups 22A15 Structure of topological semigroups 40A05 Convergence and divergence of series and sequences
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