## Varieties with tolerance and congruence extension property.(English)Zbl 0575.08001

A tolerance on an algebra is defined similarly as a congruence, only the requirement of transitivity is omitted. If a, b are two elements of an algebra $${\mathfrak A}$$, then $$T_ A(a,b)$$ is the least tolerance T on $${\mathfrak A}$$ such that (a,b)$$\in T$$; each tolerance $$T_ A(a,b)$$ is called a principal tolerance on $${\mathfrak A}$$; analogously a principal congruence on $${\mathfrak A}$$ is defined. A variety $${\mathcal V}$$ of algebras has the congruence extension property (CEP), if for any algebra $${\mathfrak A}\in {\mathcal V}$$ and any subalgebra $${\mathfrak B}$$ of $${\mathfrak A}$$ every congruence on $${\mathfrak B}$$ is the restriction of a congruence on $${\mathfrak A}$$. Analogously principal congruence extension propety (PCEP), tolerance extension property (TEP) and principal tolerance extension property (PTEP) are defined. Some conditions for varieties of algebras to have CEP, TEP, PCEP or PTEP are presented; they are expressed in terms of polynomials on algebras of such a variety. A lot of illustrating examples are shown.