Congruences, ideals and annihilators in standard QBCC-algebras
An algebra \((A,\circ, 1)\) of type \((2,0)\) is called a QBCC-algebra if it satisfies the following axioms: (1) \((x\circ y)\circ[(z\circ x)\circ(z \circ y)]= 1\), (2) \(x\circ x= 1\), (3) \(x\circ 1= x\), (4) \(1\circ x= x\). A QBCC-algebra is called standard if every subset containing 1 is a subalgebra. For \(a\in A\), a subset \(C(a)= \{x\in A\mid a\leq x\) and \(x\leq a\}\) is called the cell of \(a\), where \(x\leq y\) is defined by \(x\circ y= 1\). The authors show the following characterization theorem for the congruence lattices \(\text{Con}(A)\) of the standard QBCC-algebra \(A\) to be distributive: Theorem 2.2. \(\text{Con}(A)\) is distributive if and only if \(A\) contains at most two-element cells.
scientific article; zbMATH DE number 2143660
scientific article