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**Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories.**
*(English)*
Zbl 1062.18004

This is the reproduction of F. W. Lawvere’s famous Ph.D. thesis [“Functorial semantics of algebraic theories, Columbia University (1963)] augmented by the author’s comments after more than four decades. The summary was published [Proc. Natl. Acad. Sci. USA 50, 869–872 (1963; Zbl 0119.25901)] and this was followed by an expository paper [“Some algebraic problems in the context of functional semantics of algebraic theories”, Rep. Midwest Category Semin. 2, Lect. Notes Math. 61, 41–61 (1968; Zbl 0204.33802)]. Although the thesis itself had not been published until 2004, the ideas of the thesis have been well known and influential among category theorists. One can easily find his ideas about the categorical approach to universal algebra in such readable textbooks as B. Pareigis [“Categories and functors”, Pure Appl. Math 39 (1970; Zbl 0211.32402)]. Therefore the author’s retrospective comments on his ideas in the 1960’s, rather than the thesis itself, may presumably help the reader understand how this distinguished mathematician peeped into mathematics of the third millennium from the closing days of the second millennium.

Reviewer: Hirokazu Nishimura (Tsukuba)

### MSC:

18C10 | Theories (e.g., algebraic theories), structure, and semantics |