Balanced congruences. (Q2773036)

If \(\mathfrak{A}\) is an algebra then \(\text{Con}({\mathfrak A})\) is the set of all congruences on \({\mathfrak A}\). If \(\Phi\in \text{Con}({\mathfrak A})\), \(x\in{\mathfrak A}\), then \([x]\Phi=\{t\in{\mathfrak A}\mid (x;t)\in\Phi\}\). If \(M\subseteq{\mathfrak A}\) then \(\Theta(M)\) is the least congruence on \({\mathfrak A}\) containing the relation \(M\times M\). An algebra \({\mathfrak A}\) with two distinct nullary operations \(0\) and \(1\) is called balanced if for all \(\Phi,\Psi\in \text{Con}({\mathfrak A})\) we have \([0]\Phi=[0]\Psi\) if and only if \([1]\Phi=[1]\Psi\). A variety \({\mathfrak V}\) with two distinct nullary operations \(0\) and \(1\) is called balanced if every \({\mathfrak A}\in{\mathfrak V}\) is balanced. The paper consists of three paragraphs. Balanced congruences on bounded lattices are considered in first paragraph. Let \({\mathfrak L}=\langle L;\vee,\wedge,0,1\rangle\) be a bounded lattice with least element \(0\) and greatest element \(1\). For \(\Phi\in \text{Con}({\mathfrak L})\) put \(I=[0]\Phi\) and \(F=[1]\Phi\). We say that \(\Phi\) is balanced if \([0]\Phi=[0]\Theta(F)\) and \([1]\Phi=[1]\Theta(I)\). We say that \({\mathfrak L}\) is a \(d\)-lattice if for each \(a,b,c,d\in L\) the following holds: NEWLINE\[NEWLINE (a;0)\in\Theta(\{c;1\})\Rightarrow a\wedge c=0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE (b;1)\in\Theta(\{d;0\})\Rightarrow b\vee d=1. NEWLINE\]NEWLINE Theorem 1. Let \({\mathfrak L}\) be a \(d\)-lattice and \(\Phi\in \text{Con}({\mathfrak L})\). Then \(\Phi\) is balanced if and only if NEWLINE\[NEWLINE [0]\Phi=\{a\in L:c\wedge a=0\text{ for some }c\in[1]\Phi\} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE [1]\Phi=\{b\in L:d\vee b=1\text{ for some }d\in[0]\Phi\}. NEWLINE\]NEWLINE Balanced algebras with two nullary operations are the subject of the second paragraph. Let \(\tau\) be a type containing two distinct nullary operations denoted by \(0\) and \(1\).NEWLINENEWLINETheorem 2. Let \({\mathfrak V}\) be a variety of type \(\tau\) containing two distinct nullary operations \(0\) and \(1\). \({\mathfrak V}\) is balanced if for each \({\mathfrak A}\in{\mathfrak V}\) and every \(\Phi\in \text{Con}({\mathfrak V})\) the following property holds: \([0]\Phi\) is a singleton if and only if \([1]\Phi\) is a singleton.NEWLINENEWLINE A characterization of balanced varieties by a Mal'tsev condition is obtained in the third paragraph.