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Importation lattices

机译：进口格子

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摘要

In recent years, many works have appeared that propose order from basic fuzzy logic connectives. However, all of them assume the connectives to possess some kind of monotonicity, which succinctly implies that the underlying set is already endowed with an order. In this work, given a set P not equal empty set we define an algebra based on an implicative-type function I without assuming any order-theoretic properties, either on P or I. Terming it the importation algebra, since the law of importation becomes one of the main axioms in this algebra, we show that such algebras can impose an order on the underlying set P to obtain importation posets. In the case P = [0, 1], it is well-known that only R-implications obtained from left-continuous t-norms lead us to richer order theoretic structures. However, we show here that we can obtain new order-theoretic structures on it even from other families of fuzzy implications, in fact, even when I is not a fuzzy implication, and that one can recover the usual order on [0, 1] even from fuzzy implications that do not have the classical ordering property. We also give some sufficient conditions under which such importation posets become lattices. Finally, we present some further possible explorations. (C) 2020 Elsevier B.V. All rights reserved.
机译：近年来，许多作品似乎从基本模糊逻辑连接点提出了命令。然而，所有这些都假设联系人拥有某种单调性，简洁地意味着底层集已经赋予了订单。在这项工作中，给定一个集合不等于空集，我们基于Inclicative-Type函数I定义代数，而不假设在P或i上的任何订单理论属性。作为进口代数来称之为，因为进口法变为该代数中的一个主要公理，我们表明这种代数可以在底层设置P上施加订单，以获得进口活页塞。在情况下，P = [0,1]，众所周知，只有从左连续T-NURMS获得的r-产生的影响导致我们富有秩序的理论结构。但是，我们在这里展示了，即使来自其他模糊含义的家庭，我们也可以获得新的订单理论结构，实际上，即使我不是模糊的含义，也可以在[0,1]上恢复通常的顺序即使来自没有经典订购属性的模糊含义。我们还提供了一些足够的条件，其中这种进口活页塞成为格子。最后，我们展示了一些进一步的探险。 （c）2020 Elsevier B.v.保留所有权利。

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