Complementary Pair of Quasi-antiorders

The aims of the present paper are to introduction and investigate of notions of complementary pairs of quasi-antiorders and half-space quasi-antiorder on a given set. For a pair α and β of quasi-antiorders on a given set A we say that they are complementary pair if α ∪ β =6=A and α ∩ β = ∅. In that case, α (and β ) is called half-space on A. Assertion, if α is a half-space quasi-antiorder on A, then the induced anti-order θ on A/(α ∪ α−1) is a half-space too, is the main result of this paper

Complementary Pair of Quasi-antiorders

The aims of the present paper are to introduction and investigate of notions of complementary pairs of quasi-antiorders and half-space quasi-antiorder on a given set. For a pair $\alpha$ and $\beta$ of quasi-antiorders on a given set $A$ we say that they are complementary pair if $\alpha \cup \beta=\ne_A$ and $\alpha \cap \beta=\emptyset$. In that case, $\alpha$ (and $beta$ ) is called half-space on $A$. Assertion, if $\alpha$ is a half-space quasi-antiorder on $A$, then the induced anti-order $\theta$ on $A$ $/(\alpha\cup\alpha^{−1})$ is a half-space too, is the main result of this paper