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Pseudo-complementation on almost distributive lattices. (English) Zbl 0982.06011

An algebra \(L=(L;\vee,\wedge,0)\) of type \((2,2,0)\) is called an almost distributive lattice (ADL for short) if it satisfies the following identities: (i) \((a\vee b)\wedge c=(a\wedge c)\vee(b\wedge c)\); (ii) \(a\wedge (b \vee c)=(a\wedge b)\vee (a\wedge c)\); (iii) \((a\vee b)\wedge b=b\); (iv) \((a\vee b)\wedge a=a\); (v) \(a\vee (a\wedge b)=a\); (vi) \(0\wedge a=0\). Clearly, every distributive lattice with 0 is an ADL. It can be shown that every ADL is partially ordered by the relation \(a\leq b\) iff \(a=a\wedge b\) (or equivalently, \(a\vee b=b)\) [see U. M. Swamy and G. C. Rao, J. Austral. Math. Soc., Ser. A 31, 77-91 (1981; Zbl 0473.06008)]. Moreover, if \(a\leq b\) in \(L\), then \([a,b]\) is a bounded distributive lattice. Therefore, an ADL with the largest element 1 is a distributive lattice.
A unary operation \(a\to a^*\) on an ADL \(L\) is called a pseudocomplementation on \(L\), if it satisfies the following conditions: (j) \(a\wedge b=0\) implies \(a^*\wedge b=b\); (jj) \(a\wedge a^*=0\); (jjj) \((a\vee b)^*=a^*\wedge b^*\).
Main results: (1) Let \(L\) be an ADL with a pseudocomplementation \(\phantom{}^*\). Then the set \(B(L^{**})=\{x\in L:x=x^{**} \}\) is a Boolean algebra. (2) The class of all ADLs with pseudocomplementation is equational. (3) Having an ADL with two pseudocomplementations \(\phantom{}^*\) and \(\phantom{}^+\), then \(B(L^{**}) \cong B(L^{++})\).

MSC:

06D15 Pseudocomplemented lattices

Citations:

Zbl 0473.06008
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