Connected relation

In mathematics, a relation on a set is called connected or complete or total if it relates (or "compares") all pairs of elements of the set in one direction or the other while it is called strongly connected if it relates pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all there is a so that (see serial relation). Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty domain). A relation on a set is called when for all or, equivalently, when for all A relation with the property that for all is called . The main use of the notion of connected relation is in the context of orders, where it is used to define total, or linear, orders. In this context, the property is often not specifically named. Rather, total orders are defined as partial orders in which any two elements are comparable. Thus, is used more generally for relations that are connected or strongly connected. However, this notion of "total relation" must be distinguished from the property of being serial, which is also called total. Similarly, connected relations are sometimes called , although this, too, can lead to confusion: The universal relation is also called complete, and "complete" has several other meanings in order theory. Connected relations are also called or said to satisfy (although the more common definition of trichotomy is stronger in that of the three options must hold).

Succinct ordering and aggregation constraints in algebraic array theories

Experimental and ab initio derivation of interface stress in nanomultilayered coatings: Application to immiscible Cu/W system with variable in-plane stress

Combinatory Array Logic with Sums

Succinct ordering and aggregation constraints in algebraic array theories

Experimental and ab initio derivation of interface stress in nanomultilayered coatings: Application to immiscible Cu/W system with variable in-plane stress

Combinatory Array Logic with Sums