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On fixed points of the lower set operator
Lower subsets of an ordered semigroup form in a natural way an ordered semigroup. This lower set operator gives an analogue of the power operator already studied in semigroup theory. We present a complete description of the lower set operator applied to varieties of ordered semigroups. We also obtain large families of fixed points for this operator applied to pseudovarieties of ordered semigroups, including all examples found in the literature. This is achieved by constructing six types of inequalities that are preserved by the lower set operator. These types of inequalities are shown to be independent in a certain sense. Several applications are also presented, including the preservation of the period for a pseudovariety of ordered semigroups whose image under the lower set operator is proper
Random fixed point theorems under mild continuity assumptions
In this paper, we study the existence of the random fixed points under mild
continuity assumptions. The main theorems consider the almost lower
semicontinuous operators defined on Frechet spaces and also operators having
properties weaker than lower semicontinuity. Our results either extend or
improve corresponding ones present in literature.Comment: 15 page
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