On a theorem of Ţăndăreanu and Tudor

Abstract

For an operator domain $\mathbf{\Sigma}$, which has exactly one binary operator symbol $\sigma$, and a set $M$,  \c{T}\u{a}nd\u{a}reanu and Tudor have defined a homomorphism $f_{M}$ from the inf-semi-lattice $\mathbf{Sub}(M)$, where $\mathrm{Sub}(M)$, the underlying set of $\mathbf{Sub}(M)$, is the set of all subsets of $M$, to the inf-semi-lattice $\mathbf{Sub}_{\mathbf{\Sigma}}(\mathbf{T}_{\mathbf{\Sigma}}(M))$, where $\mathrm{Sub}_{\mathbf{\Sigma}}(\mathbf{T}_{\mathbf{\Sigma}}(M))$, the underlying set of $\mathbf{Sub}_{\mathbf{\Sigma}}(\mathbf{T}_{\mathbf{\Sigma}}(M))$, is the set of all subalgebras of the free $\mathbf{\Sigma}$-algebra $\mathbf{T}_{\mathbf{\Sigma}}(M)$ on $M$, by assigning to each $X\subseteq M$ precisely $\mathrm{T}_{\mathbf{\Sigma}}(X)$, the underlying set of the free $\mathbf{\Sigma}$-algebra $\mathbf{T}_{\mathbf{\Sigma}}(X)$ on $X$, identified to $\mathrm{Sg}_{\mathbf{T}_{\mathbf{\Sigma}}(M)}(X)$, the subalgebra of $\mathbf{T}_{\mathbf{\Sigma}}(M)$ generated by $X$. In this note we show, on the one hand, that the aforementioned homomorphisms between inf-semi-lattices are the components of a natural transformation between two suitable contravariant functors, and, on the other hand, that when the above mentioned homomorphisms are considered as order preserving mappings, they are the components of a natural transformation between two appropriate functors.