Implicitly equivalent universal algebras

The concept of implicit operation on pseudovarieties of semigroups goes back to Eilenberg and Schutzenberger [1]. The author in [2–5] generalized this concept to other classes of algebras and established a connection between these operations and positively conditional termal functions in the case of uniform local finiteness of the algebras of the class in question. In this article we put forth the concept of an implicit operation for an arbitrary universal algebra, not necessarily locally finite, and establish a connection between these operations and infinite analogs of positively conditional terms, as well as $\infty$-quasi-identities arising in the algebraic geometry of universal algebras. We also consider conditions for implicit equivalence of algebras to lattices, semilattices, and Boolean algebras.