The structure of tensor products of semilattices with zero
George Grätzer, H. Lakser, Robert W. Quackenbush · Transactions of the American Mathematical Society · 1981
If A A and B B are finite lattices, then the tensor product C C of A A and B B in the category of join semilattices with zero is a lattice again. The main result of this paper is the description of the congruence lattice of C C as the free product (in the category of bounded distributive lattices) of the congruence lattice of A A and the congruence lattice of B B . This provides us with a method of constructing finite subdirectly irreducible (resp., simple) lattices: if A A and B B are finite subdirectly irreducible (resp., simple) lattices then so is their tensor product. Another application is a result of E. T. Schmidt describing the congruence lattice of a bounded distributive extension of M 3 {M_3} .