Lawvere’s symmetric monoidal preorder, Cost.
Let \([0,\infty]\) represent the non-negative real numbers with infinity. Also take the usual notion of \(\geq\).
There is a monoidal structure for this preorder: \(\mathbf{Cost}:=([0,\infty],\geq,0,+)\)
The monoidal unit being zero means “you can get from a to a at no cost."
The product being + means “getting from a to c is at most the cost of a to b plus b to c"
The ‘at most’ above comes from the \(\geq\).