SETS AND POSETS WITH INVERSIONS

Arpad Szaz

Abstract: 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 $$ \alignat 3 x_{\blacktriangledown}&=x\land x^{\lor},& x_{\blacktriangle}&=x\land x^{\land},& x_{\blacklozenge}&=x\land x^{\lozenge};
x_{\bullet}&=x^{\lor}\land x^{\land},\quad& x_{\clubsuit}&=x^{\lor}\land x^{\lozenge},\quad& x_{\spadesuit}&=x^{\land}\land x^{\lozenge}; \endalignat $$ 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 $$ F^{\lor}(u)=F(-u), \quad F^{\land}(u)=-F(u) \quad \text{and} \quad F^{\lozenge}(u)=-F(-u) $$ for all $F\subset X\times Y$ and $u\in U$.

Classification (MSC2000): 06A06, 06A11; 06A12, 20M15

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