Description: Notes for forms [159 A] and [159 B] from Blass [1983b]
Content:
Notes for forms [159 A] and [159 B] from Blass [1983b]
A similarity type \(\tau\) is a set \(J\) of operation symbols and for
each \(j \in J\), an arity \(I_{j}\) (\(I_{j}\) indexes the argument places
of \(j\)). An algebra of type \(\tau\) is a set \(A\) and for each \(j \in
J\), an operation \(j_{A}\) from \(A^{I_{j}}\) into \(A\). The following
is a theorem of ZF. (See Blass [1983b] and
Theorem: For any similarity type \(\tau\) and set \(V\) there is an algebra \(F(V)\) of type \(\tau\) (called the \(\tau\) (or absolutely free algebra generated by \(V\)) with the property that \(V \subseteq F(V)\) and for any \(A\) of type \(\tau\) and any \(\eta : V\rightarrow A\) there is a unique homomorphism \(\alpha : F(V)\rightarrow A\) with \(\alpha(v) = v\) for all \(v \in V\).
Definition: A \(\tau\) identity is a triple \((V,w,w')\) where \(V\) is a set (of variables) and \(w\) and \(w'\) are words in the free \(\tau\) algebra \(F(V)\).
Given a similarity type \(\tau\) and a set of identities \(E\), the variety determined by \(\tau\) and \(E\) is the class of all algebras of type \(\tau\) where all the identities in \(E\) are true. \(((V,w,w')\) is true in \(A\) if for every function \(\eta : V\rightarrow A\), \(\alpha(w)= \alpha(w')\) where \(\alpha\) is the unique homomorphism extending \(\eta\).
If \(V\) is a variety and \(X\) is a set then a free algebra in the variety on \(X\) is an algebra \(A\) and a function \(\eta : X\rightarrow A\) such that for any \(A'\) in the variety, and \(\eta': X\rightarrow A'\), there is a unique homomorphism \(\alpha : A\rightarrow A'\) with \(\eta' = \alpha\eta\).
If \(X=\emptyset\) then \(A\) is called an initial algebra in the variety.
Howard-Rubin number: 32
Type: Notes on notes
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