Subdirectly irreducible semilattices with endomorphism

Abstract

In this paper we initiate an investigation into the class of meet semilattices endowed with an endomorphism. A consideration of the subdirectly irreducible algebras leads to a description of a subclass of those algebras $(S;\wedge ,k)$ in which $(S;\wedge )$ is a meet semilattice and $k$ is an endomorphism on $S$ characterised by the property $k⩾i{d}_S$. We particularly show that such an algebra is subdirectly irreducible if and only if it is a chain with one of the following forms

1. $\cdots <a_j<a_{j-1}<\cdots <a_0$;
2. $0\cdots <a_j<a_{j-1}<\cdots <a_0$
   

in which $k\left(a_j\right)=a_{j-1}$ for $j⩾1$, $k\left(0\right)=0$ and $k\left(a_0\right)=a_0$.

Keywords

• Semilattice
• endomorphism
• subdirectly irreducible

References

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