## 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
Full Text:

### References:

 [1] CHAJDA I., DUDA J.: Finitely generated relations and their applications to permutable and n-permutable varieties. Coment. Math. Univ. 23, 1982, 41-54. · Zbl 0497.08002 [2] CSKÁCY B.: Characterizations of regular varieties. Acta Sci. Math. Szeged, 31, 1970, 187-189. [3] GRÄTZER G.: Two Maľcev-type theorems in universal algebra. J. Comb. Theory, 8, 1970, 334-342. · Zbl 0194.01401 [4] WERNER H.: A Maľcev condition on admissible relations. Algebra Univ. 3, 1973, 263.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.