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)