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The variety generated by all the ordinal sums of perfect MV-chains
We present the logic BL_Chang, an axiomatic extension of BL (see P. H\'ajek -
Metamathematics of fuzzy logic - 1998, Kluwer) whose corresponding algebras
form the smallest variety containing all the ordinal sums of perfect MV-chains.
We will analyze this logic and the corresponding algebraic semantics in the
propositional and in the first-order case. As we will see, moreover, the
variety of BL_Chang-algebras will be strictly connected to the one generated by
Chang's MV-algebra (that is, the variety generated by all the perfect
MV-algebras): we will also give some new results concerning these last
structures and their logic.Comment: This is a revised version of the previous paper: the modifications
concern essentially the presentation. The scientific content is substantially
unchanged. The major variations are: Definition 2.7 has been improved.
Section 3.1 has been made more compact. A new reference, [Bus04], has been
added. There is some minor modification in Section 3.
Two isomorphism criteria for directed colimits
Using the general notions of finitely presentable and finitely generated
object introduced by Gabriel and Ulmer in 1971, we prove that, in any (locally
small) category, two sequences of finitely presentable objects and morphisms
(or two sequences of finitely generated objects and monomorphisms) have
isomorphic colimits (=direct limits) if, and only if, they are confluent. The
latter means that the two given sequences can be connected by a back-and-forth
chain of morphisms that is cofinal on each side, and commutes with the
sequences at each finite stage. In several concrete situations, analogous
isomorphism criteria are typically obtained by ad hoc arguments. The abstract
results given here can play the useful r\^ole of discerning the general from
the specific in situations of actual interest. We illustrate by applying them
to varieties of algebras, on the one hand, and to dimension groups---the
ordered of approximately finite-dimensional C*-algebras---on the other.
The first application encompasses such classical examples as Kurosh's
isomorphism criterion for countable torsion-free Abelian groups of finite rank.
The second application yields the Bratteli-Elliott Isomorphism Criterion for
dimension groups. Finally, we discuss Bratteli's original isomorphism criterion
for approximately finite-dimensional C*-algebras, and show that his result does
not follow from ours.Comment: 10 page
A note on drastic product logic
The drastic product is known to be the smallest -norm, since whenever . This -norm is not left-continuous, and hence it
does not admit a residuum. So, there are no drastic product -norm based
many-valued logics, in the sense of [EG01]. However, if we renounce standard
completeness, we can study the logic whose semantics is provided by those MTL
chains whose monoidal operation is the drastic product. This logic is called
in [NOG06]. In this note we justify the study of this
logic, which we rechristen DP (for drastic product), by means of some
interesting properties relating DP and its algebraic semantics to a weakened
law of excluded middle, to the projection operator and to
discriminator varieties. We shall show that the category of finite DP-algebras
is dually equivalent to a category whose objects are multisets of finite
chains. This duality allows us to classify all axiomatic extensions of DP, and
to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure
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