FOM: Trad.vers.Funct.Algebra # 4
walter.felscher at uni-tuebingen.de
Fri Mar 20 09:37:45 EST 1998
Traditional versus Functorial Algebra # 4
3. From equational classes to B-algebras
Given an equational class _A of algebras; let A be an
algebra which is functionally free in _A . Let E be the set
underlying A and let U be the set of basic operations of A .
Then the category B(E,U) introduced in 1.5. is a diagram B
with b_i = E^i .
Let _C(_A) be the class _C(B) of algebras of signature S
determined from B-algebras F as C_F in 2.2. I shall show
that _C(_A) coincides with _A .
The equations defining _A hold in A , hence they hold
between the operations U . This means that in B[;1] two
morphisms u, u' to b_1 are the same if they correspond to
the two terms t, t' of such equation; hence this equation
also holds in every C_F . Thus _C(_A) is a subclass of _A .
To see the converse, observe that A is functionally free,
hence the equations <t,t'> holding in A hold in every algebra
D from _A . Thus if t, t' induce distinct operations g, g'
in D then they will induce distinct operations u, u' in A .
Consequently, I can define a map F , sending the morphisms u
of B[;1] into the correspondingly induced operations on the
underlying set E of D , and extending F to the product maps
in B , I obtain F as a B-algebra with underlying set E .
It follows from the construction of C_F that this algebra
then is D .
In the particular case that _A is the class of Boolean
algebras expressed by a particular signature, the 2-element
algebra 2 is functionally free (since every algebra in _A is a
homomorphic image of a subalgebra of a power of 2 ), a so
the diagram B simply consists of the operations on 2 (and of
their products) - independent from the particular signature
and the particular defining equations for _A .
4. Algebras under a clone
The notion CL(E,U) of the clone, generated by a set U of
operations on E , was introduced in 1.5. , and above in 3.
the category B(E,U) coming from CL(E,U) was used as the
defining diagram B for the B-algebras corresponding to
those of an equational class _A where E was the underlying
set and U the set of basic operations of an algebra A
functionally free in _A .
Given any algebra A with E and U as above, CL(E,U) is called
the clone CL(A) of A ; as a bare set it is the union of the sets
underlying the algebras H(A,m).
A morphism f from a clone CL_M on a set M to a clone CL_E on
a set E shall be a map from CL_M to CL_E preserving
superposition and the projections: f (pr_M_(m,i)) = pr_E_(m,i) .
If A and B are algebras and if h is a homomorphism from A onto
B then h induces homomorphisms h_m from H(A,m) onto H(B,m),
and the union f of the h_m then is a morphism from CL(A) to
CL(B); if CL(A)=CL(M,U) and CL(B)=CL(E,W) then f maps the
basic operations U of A onto the basic operations W of B .
Choosing a fixed clone CL (instead of a diagram B ), CL-algebras
may be introduced as morphisms defined on CL (instead of
product preserving functors defined on B ). A correspondence
between classes of CL-algebras and equational classes then
will be established by the same arguments as those used for
The functorial description of equational classes fulfills
the promise to permit a coordinate-free description of these
algebraic objects. The description by Philip Hall's clones
achieves the same aim (and the example of the clone of
operations of 2 shows that in the case of Boolean algebras
it may be even nearer to the algebraic reality); its
disadvantage is that it cannot rely on an already established
terminology as that of categories and functors. Further,
the functorial description immediately generalizes to
situations in which the underlying category is not that of
bare sets (but, say, one of topological spaces).
The inherent limitations of the functorial approach are that
it captures aspects only which are expressible by equations
(and here by no means all of them); hence it can never
replace the traditional approach in the case of more general
Yet there is a further limitation, caused by our categorical
brethren themselves: many of them do not even wish to
know the methods of traditional universal algebra. There is
flowery talk such as "the equational theory ... resides in
the commutative diagrams", but when asked to exhibit
defining equations between terms for a general situation, a
proponent knowing category theory only may have a hard time
[and the situation will be even worse for the case of
algebras under a triple (monad).] It seems that for many
categorists the world only began when it were their
brethren who started to name the concepts they employ
(vide Mclarty on March 15th : "Eilenberg and MacLane created
category theory"). Such attitude is not conductive to win
appreciation outside an in-group of believers. And then
there is the categorical mafia, one of whose members refereed
a paper not on categorical methods with the conclusion
" papers such as this give category theory a bad name and
therefore should be rigidly suppressed. "
A sad world it is.
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