## 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 $p(x,x,z)=z,\quad q(z,x,y,z)=x,\quad q(p(x,y,z),x,y,z)=y.$

### MSC:

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

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