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An equation s = t between terms is said to hold universally
in an algebra just in case it is true under all assignments
of algebra elements to the variables.
Ex: Groups are the algebras of type (0,2,1) which obey the
usual group axioms universaly. Rings are the algebras of type
(0,0,2,1,2) which obey the usual ring axioms universally.
What does it mean to say that a given equation a follows from
a given set of equations S?
There are two ways to look at this: algebraically and
Under the algebraic approach, this means that a holds
universally in every algebra where S holds universally (i.e.,
every element of S holds universally).
Under the formal approach, this means that one can derive the
equation a from the set of equations S. But what is a
derivation of a from S?
Just what it means in high school algebra when one first
learns to play around with equations.
It means that there is a finite sequence of equations ending
with a, where each equation either follows from previous
equations by the transitivity and symmetry of equality, or is 4
obtained from an earlier equation by replacing variables with
terms in such a way that the equality of all of the terms
replacing the same variable have been previously proved.
THEOREM 1. a follows from S algebraically iff a follows from
COROLLARY 2. a follows from S algebraically iff a follows
from some finite subset of S algebraically.
Of course, many algebraic contexts are not strictly
equational; e.g., fields.
This suggests looking at situ-ations that are almost, but not
quite equational. E.g., a and/or some elements of S are
negations of equations. Or implications between equations. Or
implications between conjunctions of equations. Or
disjunctions of equations.
Most generally, both a and all elements of S are of the form
*) a conjunction of equations implies a disjunction of
The degenerate cases are handled in an obvious manner.
REMARK: finite sets of statements of the form *) have the
same effect as arbitrary combinations involving negation,
disjunction, conjunction, and implication.
Once we go all the way up to *), we have passed from
equational logic to what is called free variable logic.
It would be interesting to see a systematic treatment of
notions of derivation when free variable logic is approached
incrementally from equational logi...
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