Lattice tensor products. III: Congruences (Q1410414)

[For Parts I, II by the same authors see ibid. 95, 261-279 (2002; Zbl 0997.06002) and ibid. 97, 193-198 (2002; Zbl 1012.06006).] \textit{G. Grätzer} and \textit{F. Wehrung} [J. Algebra 221, 315-344 (1999; Zbl 0961.06005)] (we quote this paper as (GW)) defined the notion of lattice tensor product \(A\boxtimes B\) and showed that it can be represented as a subset \(A\langle B\rangle\) of \(B^A\). The main result of the present paper is a construction, for a finite lattice \(A\) and a bounded lattice \(B\), of an isomorphism of \(\text{Con} A\langle B\rangle\) onto \((\text{Con} A)\langle\text{Con} B\rangle\). In the paper it is remarked that this is a particular case of a stronger result proved in (GW), but that the proof given here is much simpler than the proof in (GW). Moreover, the present result generalizes a theorem of \textit{G. Grätzer, H. Lakser} and \textit{R. Quackenbush} [Trans. Am. Math. Soc. 267, 503-515 (1981; Zbl 0478.06003)].