Abstract
In this paper we mainly deal with first-order languages without equality and introduce a weak form of equality predicate, the so-called Leibniz equality. This equality is characterized algebraically by means of a natural concept of congruence; in any structure, it turns out to be the maximum congruence of the structure. We show that first-order logic without equality has two distinct complete semantics (fll semantics and reduced semantics) related by the reduction operator. The last and main part of the paper contains a series of Birkhoff-style theorems characterizing certain classes of structures defined without equality, not only full classes but also reduced ones.
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Elgueta, R. Characterization Classes Defined without Equality. Studia Logica 58, 357–394 (1997). https://doi.org/10.1023/A:1004978316495
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DOI: https://doi.org/10.1023/A:1004978316495