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On higher derivations of partially ordered sets

Annals of the Alexandru Ioan Cuza University - Mathematics, 2021

The point of this paper is to present and study the idea of higher derivations on partially ordered sets that generalize the concept of derivations on partially ordered sets. Additionally, several characterization theorems on higher derivations are introduced. Moreover, the properties of the fixed points based on the higher derivations are examined. Finally, the properties of ideals and operations related to higher derivations are studied.

On triple derivations of partially ordered sets

Ufa Mathematical Journal, 2019

In this paper, as a generalization of derivation on a partially ordered set, the notion of a triple derivation is presented and studied on a partially ordered set. We study some fundamental properties of the triple derivation on partially ordered sets. Moreover, some examples of triple derivations on a partially ordered set are given. Furthermore, it is shown that the image of an ideal under triple derivation is an ideal under some conditions. Also, the set of fixed points under triple derivation is an ideal under certain conditions. We establish a series of further results of the following nature. Let (,) be a partially ordered set. 1. If , are triple derivations on , then = if and only if Fix () = Fix (). 2. If is a triple derivation on , then, for all ∈ ;Fix () ∩ () = (()). 3. If and are two triple derivations on , then and commute. 4. If and are two triple derivations on , then if and only if =. In the end, the properties of ideals and operations related to triple derivations are examined.

Constructive completions of ordered sets, groups and fields

Annals of Pure and Applied Logic, 2005

In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, a so-called pseudo-order. The prime example is the order of the constructive real numbers. We examine two kinds of constructive completions of pseudo-orders: order completions of pseudo-orders and Cauchy completions of (non-archimedean) ordered groups and fields. It is shown how these can be predicatively defined in type theory, also when the underlying set is non-discrete. Provable choice principles, in particular a generalisation of dependent choice, are used for showing set-representability of cuts.

On generalized derivations of partially ordered sets

Communications in Mathematics, 2019

Let P be a poset and d be a derivation on P. In this research, the notion of generalized d-derivation on partially ordered sets is presented and studied. Several characterization theorems on generalized d-derivations are introduced. The properties of the fixed points based on the generalized d-derivations are examined. The properties of ideals and operations related with generalized d-derivations are studied.