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Tolerances on semilattices. (English) Zbl 0902.06018

A \(p\)-algebra is a universal algebra \((L;\lor,\land,*,0,1)\) of type \((2,2,1,0,0)\) such that \((L;\lor,\land,0,1)\) is a bounded lattice and \(*\) is the operation of pseudocomplementation (that is \(x\leq a^*\) iff \(a\land x=0\)). For \(n\geq 1\) the class of all distributive \(p\)-algebras satisfying the identity \[ (L_n):(x_1\land x_2\land\dots\land x_n)^*\lor(x_1^*\land x_2\land\dots\land x_n)^*\lor\dots\lor(x_1\land x_2\land\dots\land x^*_n)^*=1 \] is denoted by \(B_n\) (for \(n\geq 2\) the elements of \(B_n\) are called \((L_n)\)-lattices). A distributive \(p\)-algebra in which for some \(n\geq 1\) every subinterval is an \((L_n)\)-lattice is called a relative \((L_n)\)-lattice.
If \(S\) is a \(\land\)-semilattice, a tolerance on \(S\) is a reflexive and symmetric binary relation \(T\) on \(S\) which has the substitution property with respect to \(\land\); the set of all tolerances on \(S\) forms an algebraic lattice \(\text{Tol}(S)\) (with respect to set inclusion).
The aim of this paper is to prove that the tolerance lattice \(\text{Tol}(S)\) for a semilattice \(S\) is a \(p\)-algebra (an example shows that this \(p\)-algebra fails to be a relative \(p\)-algebra).

MSC:

06D15 Pseudocomplemented lattices
06A12 Semilattices
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