Notes

Description: Definitions regarding algebras from Andreka/Nemeta [1981] and Howard/Höft [1981]

Content:

Definitions from Andreka/Nemeti [1981] and Howard/Hoft [1981].

An algebra is a pair \((A,F)\) where \(A\) is a set and \(F\) is a function from an index set \(I\) such that for each \(i\in I\), there is a \(k_{i}\in \omega\) such that \(F(i)\) is a function from \(A^{k_{i}}\) into \(A\). The function \(\Delta \) with domain \(I\) defined by \(\Delta (i) = k_{i}\) is called the type of the algebra \((A,F)\).  Two algebras of the same type are similar. (Also see notes Note 32 and Note 115.)

Definition:  Let \(K\) be any class of similar algebras

  1. \(S(K)\) = the class of subalgebras of algebras in \(K\)
  2. \(H(K)\) = homomorphic images of algebras in \(K\)
  3. \(P(K)\) = direct products of non-empty families of algebras in \(K\)
  4. \(HSP\ K = H(S(P(K)))\).
  5. \(Mod\ Eq\ K\) = the class of all algebras which  are  models  of the set of equations which hold in every member of \(K\).

The fact that [0 U] is equivalent to form 0  answers  problem 31 in Grätzer [1979]

A multi-algebra is a pair \((A,F)\) where \(A\) is a set and \(F\) is a function from an index set \(I\) such that for each \(i\in I\), there is a \(k_i\in\omega\) such that \(F(i)\) is a function from \(A^{k_{i}}\) into \({\cal P}(A) - \{\emptyset\}\).  We will denote \(F(i)\) by \(f_{i}\). The function \(\Delta\)  with domain \(I\) defined by \(\Delta(i)= k_{i}\) is called the type of the  algebra. Note that this doesn't require that constants, i\. e., 0-ary operations, have to be singleton sets.

If \((A,F)\) and \((B,G)\) are algebras or multi-algebras of the same type then a function \(\phi: A \rightarrow  B\) is a homomorphism if for every \(i\in I\) and \({\bf a} = (a_{1},\ldots a_{k_{i}}) \in  A^{k_{i}}\), \(\phi (f_{i}(a_{1},\ldots a_{k_{i}})) = g_{i}(\phi (a_{1}),\ldots \phi (a_{k_i})).\) The function \(\phi \) is an isomorphism between multi-algebras if \(\phi\) and \(\phi^{-1}\)  are bijective homomorphisms.

Definition: Suppose that \((A,F)\) is a multi-algebra of type \(\Delta \)  and \((B,G)\) is an algebra of type \(\Delta \). Let \(E\) be an equivalence relation on \(B\) and let \(C/E\)  be  the set of equivalence classes, \((C/E,G/E)\) is the multi-algebra of type \(\Delta \) defined by \[ ( G/E)(i)([a_{1}],\ldots [a_{k_{i}}]) =\left\{[b]:b = g_{i}(c_{1},\ldots c_{k_{i}}) \wedge  [c_{j}] = [a_{j}] \hbox{ for }j = 1,\ldots ,k_{i} \right\}. \] Here we have used \([a]\) to denote the \(E\) equivalence class of \(a \in C\).

Howard-Rubin number: 50

Type: Definitions

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