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A "✓" indicates that the column property is required in the row definition. For example, the definition of an equivalence relation requires it to be symmetric. All definitions tacitly require transitivity and reflexivity.
In mathematics, a homogeneous relation is called a connex relation, or a relation having the property of connexity, if it relates all pairs of elements in some way. More formally, the homogeneous relation R over a set X is connex when
A homogeneous relation is called a semiconnex relation, or a relation having the property of semiconnexity, if it relates all pairs of distinct elements in some way. More formally, the homogeneous relation R over a set X is semiconnex when
Several authors define only the semiconnex property, and call it connex rather than semiconnex.
The connex properties originated from order theory: if a partial order is also a connex relation, then it is a total order. Therefore, in older sources, a connex relation was said to have the totality property; however, this terminology is disadvantageous as it may lead to confusion with, e.g., the unrelated notion of right-totality, also known as surjectivity. Some authors call the connex property of a relation completeness.
Felix Brandt; Markus Brill; Paul Harrenstein (2016). "Tournament Solutions"(PDF). In Felix Brandt; Vincent Conitzer; Ulle Endriss; Jérôme Lang; Ariel D. Procaccia (eds.). Handbook of Computational Social Choice. Cambridge University Press. ISBN978-1-107-06043-2. Archived(PDF) from the original on 8 Dec 2017. Retrieved 2019. Page 59, footnote 1.
^defined formally by vEw if a graph edge leads from vertice v to vertice w
^For the only if direction, both properties follow trivially. — For the if direction: when x?y, then xRy ? yRx follows from the semiconnex property; when x=y, even xRy follows from reflexivity.