Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997 Association of Mizar Users

## Birkhoff Theorem for Many Sorted Algebras

Artur Kornilowicz
Warsaw University, Bialystok

### Summary.

\newcommand \pred[1]{${\cal P} #1$} In this article Birkhoff Variety Theorem for many sorted algebras is proved. A class of algebras is represented by predicate \pred{}. Notation \pred{[A]}, where $A$ is an algebra, means that $A$ is in class \pred{}. All algebras in our class are many sorted over many sorted signature $S$. The properties of varieties: \begin{itemize} \itemsep-3pt \item a class \pred{ } of algebras is abstract \item a class \pred{ } of algebras is closed under subalgebras \item a class \pred{ } of algebras is closed under congruences \item a class \pred{ } of algebras is closed under products \end{itemize} are published in this paper as: \begin{itemize} \itemsep-3pt \item for all non-empty algebras $A$, $B$ over $S$ such that $A$ and $B$ are\_isomorphic and \pred{[A]} holds \pred{[B]} \item for every non-empty algebra $A$ over $S$ and for strict non-empty subalgebra $B$ of $A$ such that \pred{[A]} holds \pred{[B]} \item for every non-empty algebra $A$ over $S$ and for every congruence $R$ of $A$ such that \pred{[A]} holds \pred{[A\slash R]} \item Let $I$ be a set and $F$ be an algebra family of $I$ over ${\cal A}.$ Suppose that for every set $i$ such that $i \in I$ there exists an algebra $A$ over ${\cal A}$ such that $A = F(i)$ and ${\cal P}[A]$. Then${\cal P}[\prod F]$. \end{itemize} This paper is formalization of parts of [21].

#### MML Identifier: BIRKHOFF

The terminology and notation used in this paper have been introduced in the following articles [16] [5] [20] [19] [14] [22] [3] [23] [4] [1] [17] [10] [18] [2] [8] [15] [13] [11] [12] [9] [6] [7]

Contents (PDF format)

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