Multi-relation Agassiz sums of algebras

Ludovico Fusco

Authors: Ludovico Fusco and Francesco Paoli

Płonka sums are a simple yet powerful technique for building new algebras out of semilattice direct systems of algebras of a fixed type. First introduced by J. Płonka [2] in 1967, this construction immediately proved to be essential for the representation of algebras in regular varieties [3]. The aptness of Płonka sums to this purpose can be seen, at the same time, as a shortcoming, since this construction is of little avail outside this particular domain of application.

Remarkably, however, the literature teems with ‘Płonka-type’ constructions used for the representation of algebras in irregular varieties, or in varieties having certain properties stronger than regularity. All these constructions bear striking similarities to Płonka sums, but differ from them in some important respects. Examples include the representation of Polin algebras [4], T. Katriňák and J. Guričan’s representation of pseudocomplemented semilattices [1], or the recently introduced De Morgan-Pł,onka sums [5].

We aim at finding a convenient umbrella under which all these constructions can be subsumed. We introduce a multi-relation variant of G. Grätzer and J. Sichler’s sums over Agassiz systems of algebras (Agassiz sums) [6] that encompasses Płonka sums as a special case. We prove that the above-mentioned representations of Polin algebras and pseudocomplemented semilattices, as well as De Morgan-Płonka sums, can be recast in terms of sums over appropriate bi-relation Agassiz systems. Finally, we investigate the problem as to which identities are preserved by the construction.

Bibliography

1. Katriňák, T., Guričan, J.,On a new construction of pseudocomplemented semilattices,Algebra Universalis,vol. LXXXII (2021), article no. 54.
2. Płonka, J.,On a method of construction of abstract algebras,Fundamenta Mathematicae,vol. LXI (1967), no. II, pp. 183–189.
3. —, On equational classes of abstract algebras defined by regular equations,Fundamenta Mathematicae,vol. LXIV (1969), no. II, pp. 241–247.
4. Polin, S.V.,Identities in congruence lattices of universal algebras,Matematicheskie Zametki,vol. XXII (1977), no. III, pp. 443–451.
5. Randriamahazaka, T.,De Morgan-Płonka sums,Studia Logica,forthcoming.
6. Grätzer, G., Sichler, J.,Agassiz sums of algebras,Colloquium Mathematicum,vol. XXX (1974), no. I, pp. 57–59.

 Overview  Program