indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
This is about Birkhoff’s variety or HSP theorem in universal algebra, related to equational logic?. While it is sometimes referred to simply as Birkhoff’s theorem beware of other well know Birkhoff's theorems.
(Birkhoff’s HSP theorem)
Given a language $L$ generated by a set of (single-sorted) finitary operations, and a class $C$ of structures for $L$. Then $C$ is the class of models for a set of universally quantified equations between terms of $L$ (a Lawvere theory) if and only if
In a more classical wording/terminology it is phrased as
(Birkhoff’s variety theorem) A class of algebras of the same signature is a variety of algebras iff it is closed under homomorphic images, subalgebras and arbitrary (small) products.
See also at Lawvere theory – Characterization of examples
Here “closed under homomorphic images” means that if $A$ and $B$ are structures in the class, and $\phi \colon A \to B$ is a homomorphism between them, then also its image $im(\phi) \hookrightarrow B$ is an element of the class.
The first-order analogue of HSP (theorem ) is the characterization (see e.g. Chang and Keisler’s original text (Chang-Keisler 66) on continuous model theory) of elementary classes of structures of structures: they’re precisely those closed under elementary substructures, elementary embeddings, ultraproducts, and ultraroots (if an ultrapower of something is in your class, that something was in your class.)
The original reference is
There is a textbook treatment in the monograph
Chang, Keisler Continuous Model Theory, Princeton University Press (1966) [ISBN: 9780691079295]
Michael Barr, HSP type theorems in the category of posets, in: Proc. 7th International Conf. Mathematical Foundation of Programming Language Semantics, Lecture Notes in Computer Science 598 (1992) 221–234 [doi:10.1007/3-540-55511-0_11, pdf]
Michael Barr, Functorial semantics and HSP type theorems, Algebra Universalis 31 (1994) 223–251 [doi:10.1007/BF01236519, pdf, pdf]
Michael Barr, HSP subcategories of Eilenberg-Moore algebras, Theory Appl. Categories 10 18 (2002), 461–468 [tac:10-18]
Robert Goldblatt, What is the coalgebraic analogue of Birkhoff’s variety theorem?, Theoretical Computer Science 266:1–2 (2001) 853–886 doi
Some Birkhoff’s variety-style theorems in a categorical setup for equational logic play role in
See also:
Last revised on August 30, 2024 at 09:49:42. See the history of this page for a list of all contributions to it.