On a theorem of ndreanu and Tudor

摘要

For an operator domain $mathbf{Sigma}$, which has exactly one binary operator symbol $sigma$, and a set $M$, c{T}u{a}ndu{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 $Xsubseteq M$ 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.