Sets and Posets with Inversions

Árpád Száz

Publications de l'Institut Mathématique, _N_S_90 (2011) no. 104, p. 111

Résumé

We investigate unary operations $\lor$, $\land$ and $\lozenge$ on a set $X$ satisfying $x=x^{\lor\lor}=x^{\land\land}$ and $x^{\lozenge}=x^{\lor\land}=x^{\land\lor}$ for all $x\in X$. Moreover, if in particular $X$ is a meet-semilattice, then we also investigate the operations defined by $ lignat 3 x_{ḻacktriangledown}=xand x^{or}, x_{ḻacktriangle}=xand x^{and}, x_{ḻacklozenge}=xand x^{ozenge}; x_{\bullet}=x^{or}and x^{and},\quad x_{ļubsuit}=x^{or}and x^{ozenge},\quad x_{padesuit}=x^{and}and x^{ozenge}; \endalignat $ and $x_{\bigstar}=x\land x^{\lor}\land x^{\land}\land x^{\lozenge}$ for all $x\in X$. Our prime example for this is the set-lattice $\Cal{P}(U,V)$ of all relations on one group $U$ to another $V$ equipped with the operations defined such that $ F^{or}(u)=F(-u), \quad F^{and}(u)=-F(u) \quad ext{and} \quad F^{ozenge}(u)=-F(-u) $ for all $F\subset X\times Y$ and $u\in U$.

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Classification : 06A06 06A11 06A12 20M15