25 research outputs found
Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond
Ruitenburg\u2019s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ulti- mately periodic if f fixes all the generators but one. More precisely, there is N 65 0 such that f^N+2 = f^N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators
Heyting frames and Esakia duality
We introduce the category of Heyting frames and show that it is equivalent to
the category of Heyting algebras and dually equivalent to the category of
Esakia spaces. This provides a frame-theoretic perspective on Esakia duality
for Heyting algebras. We also generalize these results to the setting of
Brouwerian algebras and Brouwerian semilattices by introducing the
corresponding categories of Brouwerian frames and extending the above
equivalences and dual equivalences. This provides a frame-theoretic perspective
on generalized Esakia duality for Brouwerian algebras and Brouwerian
semilattices
The Structure of Residuated Lattices
A residuated lattice is an ordered algebraic structure [formula] such that is a lattice, is a monoid, and \ and / are binary operations for which the equivalences [formula] hold for all a,b,c β L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as dividing on the right by b and dividing on the left by a. The class of all residuated lattices is denoted by ββ The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers [15, 16, 45β48] and also by Krull in [33]. Since that time, there has been substantial research regarding some specific classes of residuated structures, see for example [1, 9, 26] and [38], but we believe that this is the first time that a general structural theory has been established for the class ββ as a whole. In particular, we develop the notion of a normal subalgebra and show that ββ is an ideal variety in the sense that it is an equational class in which congruences correspond to normal subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety ββ[sup C] that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members (along the lines of the decompositions provided in [27]