Narrow Results

The basic properties of distributivity and deletion of pure and o-substitution are investigated. The obtained results are applied to show preservation of recognizability in a number of interesting cases. It is proved that linear and recognizable tree series are closed under o-substitution provided that the underlying semiring is commutative, continuous, and additively idempotent. It is known that, in general, pure substitution does not preserve recognizability (not even for linear target tree series), but it is shown that recognizable linear probability distributions (represented as tree series) are closed under pure substitution.

We investigate a class of non-involutive solutions of the Yang–Baxter equation which generalize derived (self-distributive) solutions. In particular, we study generalized multipermutation solutions in this class. We show that the Yang–Baxter (permutation) groups of such solutions are nilpotent. We formulate the results in the language of biracks which allows us to apply universal algebra tools.

A characterization of all idempotent uninorms satisfying the distributive property is given. The special cases of left-continuous and right-continuous idempotent uninorms are presented separately and it is also proved that all idempotent uninorms are autodistributive. Moreover, all distributive pairs of idempotent uninorms (pairs U₁, U₂ such that U_l is distributive over U₂ and U₂ is distributive over U₁) are also characterized.

The problem of distributivity was posed many years ago and it has been investigated for families of certain operations, such as t-norms, t-conorms, uninorms and nullnorms. In this paper, we continue to investigate the same topic as the above by focusing on semi-nullnorms and semi-t-operators, which are generalization of nullnorms and t-operators by omitting commutativity and associativity, and commutativity, respectively. The obtained results are not only the full characterization of this kind of distributivity equation, but also extensions of distributivity for nullnorms and for semi-nullnorms. Moreover, this paper finishes the full characterization of distributivity between semi-nullnorms and semi-t-operators together with Refs. 6 and 20.

We study some properties of De Morgan triplets. Firstly, we introduce submodular De Morgan triplets and we study its relationships with subdistributive ones. Moreover, we characterize them both in the strict and non strict archimedean cases. Secondly, we introduce the concepts of modularity, distributivity and (S, T)-distributivity degrees and we give some general results. Afterwards we apply these concepts to two particular cases, Lukasiewicz triplets and a kind of strict De Morgan triplets (Product triplets). For the Lukasiewicz ones we prove that all three degrees range over (0, 1/2]. For Product triplets, a recipe to calculate these degrees is given. In particular we present examples with distributivity degree r for all r ∈ (0, 1) and examples with (S, T)-distributivity and modularity degrees taking all values in (0, 3/4].