### $\U0001d504(1,1)$ can be strongly embedded into category of semigroups

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In this paper we introduce the concept of an $(L,\varphi )$-representation of an algebra $A$ which is a common generalization of subdirect, full subdirect and weak direct representation of $A$. Here we characterize such representations in terms of congruence relations.

Let $\tau $ be a type of algebras. A valuation of terms of type $\tau $ is a function $v$ assigning to each term $t$ of type $\tau $ a value $v\left(t\right)\ge 0$. For $k\ge 1$, an identity $s\approx t$ of type $\tau $ is said to be $k$-normal (with respect to valuation $v$) if either $s=t$ or both $s$ and $t$ have value $\ge k$. Taking $k=1$ with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called $k$-normal (with respect to the valuation $v$) if all its identities are $k$-normal. For any variety $V$, there is a least...

We prove here an Eilenberg type theorem: the so-called conjunctive varieties of rational languages correspond to the pseudovarieties of finite semilattice-ordered monoids. Taking complements of members of a conjunctive variety of languages we get a so-called disjunctive variety. We present here a non-trivial example of such a variety together with an equational characterization of the corresponding pseudovariety.