The semi-ring described at the end on Z_2 = {0,1} with the 'interesting' addition function where 1+1=1 might be more familiar here as Boolean logic, where:

0 -> False

1 -> True

x -> AND

+ -> OR

(a AND b)=True iff a=True and b=True; (axb)=1 iff a=1 and b=1

(a OR b)=False iff a=False and b=False; (a+b)=0 iff a=0 and b=0