Is there any successful approach to algebraization of set theory other than the category-theoretical approach?

Question

The approach towards algebraization of set theory which started to be developed in 1988 by André Joyal and Ieke Moerdijk was presented in their book titled "Algebraic Set Theory" which appeared in 1995. This approach is sometimes called "category-theoretical" since it is based on ideas and terms of category theory. However, I read that other mathematicians (e.g. Alfred Tarski, Dana Scott) also worked on algebraization of set theory and they did not use the language of category theory. Was any of such approaches successful enough to result in formulation of a set theory in terms other than category theoretic terms?

Answer 1

Tarski's ideas about algebraization of set theory (and its underlying logic) were developed in a book co-authored with Steven Givant. The MathSciNet reference is

MR0920815 Tarski, Alfred; Givant, Steven A formalization of set theory without variables. American Mathematical Society Colloquium Publications, 41. American Mathematical Society, Providence, RI, 1987. xxii+318 pp. ISBN: 0-8218-1041-3