Does someone know other examples of $K$-semigroups? In our work on projective representations we need use the following object:
Let  $K$ be a field. By a $K$-semigroup  we mean a semigroup $S$ with $0$ and a map $K \times S \to S$ such that
$\alpha (\beta x) = (\alpha \beta )x,$ $\alpha (xy) = (\alpha x ) y = x (\alpha y),$ $1_K x =x$ for any
$\alpha , \beta \in K, x,y \in S$ and the zero of $K$ multiplied by any
element of $S$ is $0 \in S.$ 
Of course  K-algebras considered as multiplicative semigroups are K-semigroups. 
I wonder if someone met other K-semigroups in his algebraic practice.
Thank you in advance.

Why do you assume that $K$ is a field? As for the definition, it could be any semigroup with $0$. What you call a $K$-semigroup is an action of $K$ on a semigroup with $0$ in the general sense of actions in monoidal categories.
For fixed $K$, the $K$-semigroups constitute a variety in the sense of universal algebra. Hence there are lots of constructions, for example quotients by congruence relations, generated substructures, free structures, limits, colimits. This gives lots of specific examples of $K$-semigroups.
For example, the free $K$-semigroup generated by an object $x$ consists of $0$ and elements of the form $(\alpha_1 \cdot x^{k_1}) \cdot \dotsc \cdot (\alpha_n \cdot x^{k_n})$, where $\alpha_i \in K \setminus \{0,1\}$ and $k_i \geq 1$.
