4 research outputs found
Epimorphisms, definability and cardinalities
We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most s non-logical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m+s+âľ0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable âbridge theoremsâ, matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic ⢠with suitable infinitary definability properties of â˘, while not making the standard but awkward assumption that ⢠comes furnished with a proper class of variables.The European Unionâs Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 689176 (project âSyntax Meets Semantics: Methods, Interactions, and Connections in Substructural logicsâ). The first author was also supported by the Project GA17-04630S of the Czech Science Foundation (GAÄR). The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.http://link.springer.com/journal/112252020-02-07hj2019Mathematics and Applied Mathematic
Epimorphism surjectivity in varieties of Heyting algebras
It was shown recently that epimorphisms need not be surjective in a variety K
of Heyting algebras, but only one counter-example was exhibited in the
literature until now. Here, a continuum of such examples is identified, viz.
the variety generated by the Rieger-Nishimura lattice, and all of its (locally
finite) subvarieties that contain the original counter-example K. It is known
that, whenever a variety of Heyting algebras has finite depth, then it has
surjective epimorphisms. In contrast, we show that for every integer n greater
or equal than 2, the variety of all Heyting algebras of width at most n has a
non-surjective epimorphism. Within the so-called Kuznetsov-Gerciu variety
(i.e., the variety generated by finite linear sums of one-generated Heyting
algebras), we describe exactly the subvarieties that have surjective
epimorphisms. This yields new positive examples, and an alternative proof of
epimorphism surjectivity for all varieties of Goedel algebras. The results
settle natural questions about Beth-style definability for a range of
intermediate logics
Idempotent residuated structures : some category equivalences and their applications
This paper concerns residuated lattice-ordered idempotent commutative
monoids that are subdirect products of chains. An algebra of this
kind is a generalized Sugihara monoid (GSM) if it is generated by the lower
bounds of the monoid identity; it is a Sugihara monoid if it has a compatible
involution :. Our main theorem establishes a category equivalence
between GSMs and relative Stone algebras with a nucleus (i.e., a closure
operator preserving the lattice operations). An analogous result is obtained
for Sugihara monoids. Among other applications, it is shown that Sugihara
monoids are strongly amalgamable, and that the relevance logic RMt has
the projective Beth de nability property for deduction.http://www.ams.org//journals/tran/hb201
Algebraic characterizations of various Beth definability properties
In this paper it will be shown that the Beth definability property corresponds to surjectiveness of epimorphisms in abstract algebraic logic. This generalizes a result by I. N'emeti (cf. [HMT85, Theorem 5.6.10]). Moreover, an equally general characterization of the weak Beth property will be given. This gives a solution to Problem 14 in [Sai90]. Finally, the characterization of the projective Beth property for varieties of modal algebras by L. Maksimova (see [Mak97]) will be shown to hold for the larger class of semantically algebraizable logics. 1 Introduction Abstract algebraic logic is typically concerned with equivalence theorems of the following kind, Logic L has property P () Alg(L) has property alg(P ), where Alg(L) denotes the class of algebras associated to the logic L in some canonical way, on which we come to speak later. The main motivation for making these efforts is that it shows us that algebra and logic are, so to speak, nothing but two sides of the same coin; they stu..