Construction of commutative and associative operations by paving

Paving is a method for constructing new operations from a given one. We will show that this method can be used to construct associative, commutative and monotone operations from particular given operations (from basic ‘paving stones’). We will discuss properties of the resulting operations by considering different cases of the ‘paving stones’ and the starting position of paving. Finally, we will discuss the case when the basic ‘paving stone’ is a generated operation. We show that in this case we get by paving also a generated operation, just the generator is a two-place function. We show also an example of a non-representable uninorm which is strictly increasing in both variables on the open unit square.