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On the coextension of cut-continuous pomonoids. (English) Zbl 1476.06008

A partially ordered monoid, or pomonoid, is a monoid endowed with a compatible partial order, a pomonoid ia called cut-continuous if the product \(\cdot\) is separately cut-continuous, that is, the sets \(\{z : y\cdot z\leq x\}\) and \(\{z : z\cdot y \leq x\}\) are cuts for any \(x, y\in L.\) cut-continuous pomonoids are generalization of residuated posets. In the case of a total order, the condition of cut-continuity means that multiplication distributes over existing suprema. Morphisms between cut-continuous pomonoids can be chosen either in analogy with unital quantales or with residuated lattices. Under the assumption of commutativity and integrality, congruences on cut-continuous pomonoids are induced by filters, in the same way as known for residuated lattices. The authors give the construction of coextensions, given cut-continuous pomonoids \(K\) and \(C\), how to determine the cut-continuous pomonoids \(L\) such that \(C\) is a filter of \(L\) and the quotient of \(L\) induced by \(C\) is isomorphic to \(K.\) The authors are in particular concerned with tensor products of modules over cut-continuous pomonoids. Using results of M. Erné and J. Picado on closure spaces [Algebra Univers. 78, No. 4, 461–487 (2017; Zbl 1420.06025)], the authors show that such tensor products exist. An application is the construction of residuated structures related to fuzzy logics, in particular left-continuous t-norms.

MSC:

06F05 Ordered semigroups and monoids
20M10 General structure theory for semigroups

Citations:

Zbl 1420.06025
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References:

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