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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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