Let $L$ be a lattice. In this paper, corresponding to a given congruence relation $\Theta$ of $L$, a congruence relation ${\Psi }_{\Theta }$ on $CS\left(L\right)$ is defined and it is proved that 1. $CS(L/\Theta )$ is isomorphic to $CS\left(L\right)/{\Psi }_{\Theta }$; 2. $L/\Theta$ and $CS\left(L\right)/{\Psi }_{\Theta }$ are in the same equational class; 3. if $\Theta$ is representable in $L$, then so is ${\Psi }_{\Theta }$ in $CS\left(L\right)$.

The primary aim of this article is to put forward new classes of uni-nullnorms on certain classes of bounded lattices via closure (interior) operators. Due to the new classes of uninorms combining both a t-norm $T$ and a t-conorm $S$ by various kinds of closure operators or interior operators, the relationships and properties among the same class of uninorms on $L$, we obtain new classes of uni-nullnorms on $L$ via closure (interior) operators. The constructions of uni-nullnorms on some certain classes...

In this note, we point out that Theorem 3.1 as well as Theorem 3.5 in G. D. Çaylı and F. Karaçal (Kybernetika 53 (2017), 394-417) contains a superfluous condition. We have also generalized them by using closure (interior, resp.) operators.

In the study, we introduce the definition of a locally internal uninorm on an arbitrary bounded lattice $L$. We examine some properties of an idempotent and locally internal uninorm on an arbitrary bounded latice $L$, and investigate relationship between these operators. Moreover, some illustrative examples are added to show the connection between idempotent and locally internal uninorm.