[21W]Given two ordered sets \((X,≤_ X)\) and \((Y,≤_ Y)\), with \(X,Y\) disjoint, the concatenation of \(X\) with \(Y\) is obtained defining \(Z=X∪ Y\) and providing it with the ordering \(≤_ Z\) given by:
if \(z_ 1,z_ 2∈ X\) then \(z_ 1≤_ Z z_ 2\) if and only if \(z_ 1≤_ X z_ 2\);
if \(z_ 1,z_ 2∈ Y\) then \(z_ 1≤_ Z z_ 2\) if and only if \(z_ 1≤_ Y z_ 2\);
If \(z_ 1∈ X\) and \(z_ 2∈ Y\) then you always have \(z_ 1≤_ Z z_ 2\).
This operation is sometimes denoted by the notation \(Z = X⧺ Y\).
If the sets are not disjoint, we can replace them with disjoint sets defined by \(\tilde X=\{ 0\} × X\) and \(\tilde Y=\{ 1\} × Y\), then we may "copy" the respective orders, and finally we can perform the concatenation of \(\tilde X\) and \(\tilde Y\).