The second isomorphism theorem on ordered set under antiorders

Daniel A. Romano

Kragujevac Journal of Mathematics, Tome 30 (2007) no. 1.

Résumé

In this article we give two new characteristics of quasi-antiorder relation on ordered set under antiorder.\\ The new results in this article is co-called the second isomorphism theorem on ordered sets under antiorders: Let $(X,=,\neq,\alpha)$ be an ordered set under antiorder $\alpha$, $\rho$ and $\sigma$ quasi-antiorders on $X$ such that $\sigma\subseteq\rho$. Then the relation $\sigma/\rho$, defined by $$\sigma/\rho=\{(x(\rho\cup\rho^{-1}),y(\rho\cup\rho^{-1})\in X/(\rho\cup\rho^{-1})\times X/(\rho\cup\rho^{-1}):(x,y)\in\sigma\},$$ is a quasi-antiorder on $X/(\rho\cup\rho^{-1})$ and $(X/(\rho\cup\rho^{-1}))/((\sigma/\rho)\cup(\sigma/\rho)^{-1})\cong X/(\sigma\cup\sigma^{-1})$ holds.\\ Let $\mathbf{A}=\{\tau: \tau$ is quasi-antiorder on $X$ such that $\tau\subset\sigma\}$. Let $\mathbf{B}$ be the family of all quasi-antiorder on $X/q$, where $q=\sigma\cup\sigma^{-1}$. We shall give connection between families $\mathbf{A}$ and $\mathbf{B}$. For $\tau\in\mathbf{A}$, we define a relation $\psi(\tau)=\{(aq,bq)\in X/q\times X/q:(a,b)\in\tau\}$. The mapping $\psi:\mathbf{A}\rightarrow\mathbf{B}$ is strongly extensional, injective and surjective mapping from $\mathbf{A}$ onto $\mathbf{B}$ and for $\tau, \mu\in\mathbf{A}$ we have $\tau\subseteq\mu$ if and only if $\psi(\tau)\subseteq\psi(\mu)$.