Anatolii V. Zhuchok – The least dimonoid congruence on the free -nilpotent -dimonoid

Luhansk Taras Shevchenko National University (Starobilsk, Ukraine)
The -dimonoids [2] are a generalization of dimonoids [1] while every 0-dialgebra with associative operations [3] is a linear analog of a -dimonoid. Free -nilpotent -dimonoids were constructed in [4].

Let be an arbitrary nonempty set, and let be an arbitrary word over . For every , the number of occurrences of the element in is denoted by . If is a congruence on a -dimonoid such that is a dimonoid, we say that is a dimonoid congruence. If is a congruence on a -dimonoid such that operations of coincide, we say that is a semigroup congruence.

Theorem. Let be the free -nilpotent -dimonoid, .

(i) Define a relation on by

if and only if one of the following conditions holds:

(1) and , ;

(2) .

Then is the least dimonoid congruence on .

(ii) Define a relation on by

if and only if one of the following conditions holds:

(1) and ;

(2) .

Then is the least semigroup congruence on .

References

[1]
Loday, J.-L.: Dialgebras. In: Dialgebras and related operads: Lect. Notes Math., vol. 1763, Berlin: Springer-Verlag, 7–66 (2001)
[2]
Movsisyan, Y., Davidov, S., Safaryan, M.: Construction of free -dimonoids. Algebra Discrete Math. 18, no. 1, 138–148 (2014)
[3]
Pozhidaev, A.P.: 0-dialgebras with bar-unity and nonassociative Rota-Baxter algebras. Sib. Math. J. 50, no. 6, 1070–1080 (2009)
[4]
Zhuchok, Yul. V.: On one class of algebras. Algebra Discrete Math. 18, no. 2, 306–320 (2014)