Paolo Lipparini
Reports on Mathematical Logic, Number 59, 2024, pp. 79-95
https://doi.org/10.4467/20842589RM.24.001.20699We show that a variety with J´onsson terms t1, . . . , tn–1 has directed J´onsson terms d1, . . . , dn–1 for the same value of the indices, solving a problem raised by Kazda et al. Refined results are obtained for locally finite varieties.
Paolo Lipparini
Reports on Mathematical Logic, Number 57, 2022, pp. 31-43
https://doi.org/10.4467/20842589RM.22.002.16659Contrary to the finitary case, the set Γ(A) of all the non-generators of an infinitary algebra A is not necessarily a subalgebra of A. We show that the phenomenon is ubiquitous: every algebra with at least one infinitary operation can be embedded into some algebra B such that Γ(B) is not a subalgebra of B. As far as expansions are concerned, there are examples of infinite algebras A such that in every expansion B of A the set Γ(B) is a subalgebra of B. However, under relatively weak assumptions on A, it is possible to get some expansion B of A such that Γ(B) fails to be a subalgebra of B.