Transferable principal congruences and regular algebras. (English) Zbl 0601.08004

An algebra \(\mathfrak A\) has transferable principal congruences (briefly TPC) if for any elements \(a,b,c\in\mathfrak A\) there is an element \(d\in\mathfrak A\) such that \(\Theta (a,b)=\Theta (c,d)\). A variety \(\mathcal V\) has TPC if each \(\mathfrak A\in\mathcal V\) has TPC.
Theorem 1. For a variety \(\mathcal V\), the following conditions are equivalent: (1) \(\mathcal V\) has TPC: (2) there is a ternary polynomial \(p\) and 5-ary polynomials \(q_ 1,\ldots,q_ m\) such that \[ p(x,x,z)=z,\quad q_ 1(z,p(x,y,z),x,y,z)=x,\quad q_ m(p(x,y,z),z,x,y,z)=y, \]
\[ q_{j- 1}(p(x,y,z),z,x,y,z)=q_ j(z,p(x,y,z),x,y,z)\text{ for } j=2,\ldots,m; \] (3) there is a ternary polynomial \(p\) such that \(p(x,y,z)=z\) iff \(x=y\).
Theorem 2. For a variety \(\mathcal V\), the following conditions are equivalent: (1) \(\mathcal V\) is permutable and has TPC; (2) there is a ternary polynomial \(p\) and a 4-ary polynomial \(q\) such that \[ p(x,x,z)=z,\quad q(z,x,y,z)=x,\quad q(p(x,y,z),x,y,z)=y. \]


08B05 Equational logic, Mal’tsev conditions
08A30 Subalgebras, congruence relations
08A40 Operations and polynomials in algebraic structures, primal algebras
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