
doi: 10.7151/dmgaa.1031
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: \[ (a;0)\in\Theta(\{c;1\})\Rightarrow a\wedge c=0, \] \[ (b;1)\in\Theta(\{d;0\})\Rightarrow b\vee d=1. \] Theorem 1. Let \({\mathfrak L}\) be a \(d\)-lattice and \(\Phi\in \text{Con}({\mathfrak L})\). Then \(\Phi\) is balanced if and only if \[ [0]\Phi=\{a\in L:c\wedge a=0\text{ for some }c\in[1]\Phi\} \] and \[ [1]\Phi=\{b\in L:d\vee b=1\text{ for some }d\in[0]\Phi\}. \] 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 \(1\). Theorem 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. A characterization of balanced varieties by a Mal'tsev condition is obtained in the third paragraph.
Mal'tsev condition, Lattice ideals, congruence relations, balanced algebra, balanced congruence, Equational logic, Mal'tsev conditions, Subalgebras, congruence relations, balanced variety
Mal'tsev condition, Lattice ideals, congruence relations, balanced algebra, balanced congruence, Equational logic, Mal'tsev conditions, Subalgebras, congruence relations, balanced variety
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