Let L ∗ M denote the coproduct of the bounded distributive lattices L and M . At the 1981 Banff Conference on Ordered Sets, the following question was posed: What is the largest class L of finite distributive lattices such that, for every non-trivial Boolean lattice B and every L ∈ L, B ∗ L = B ∗ L′ implies L = L′? In this note, the problem is solved.

for any x 1 ( . . . , x,, GX. A theorem of Aolci and Rolewicz (see [18]) asserts that if in (1.3) C = 2~\ then X is p-normable. We can then equivalently re-norm X so that in (1.4) JB = 1. If in addition X is a vector lattice and ||x||<||y|| whenever |x|<|y| we say that X is a quasi-Banach lattice. As in the case of Banach lattices [13] we may make the following definitions. We shall say that X ...

In this article, we prove that a lattice of minimum covolume in a simple Lie group over a positive characteristic local field is non-uniform if the Weil’s conjecture on Tamagawa numbers [Wei61] holds. This, in part, answers Lubotzky’s conjecture [Lub91]

Yasir BASHIR1,∗, Faisal NADEEM, Ayesha SHABBIR Faculty of Mathematics, COMSATS Institute of Information Technology Wah Campus, Wah Cantt, Pakistan Faculty of Mathematics, COMSATS Institute of Information Technology Lahore Campus, Lahore, Pakistan Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore, Pakistan University College of Engineering, Sciences and Te...

Omjyoti Dutta, Mariusz Gajda, Philipp Hauke, Maciej Lewenstein, Dirk-Sören Lühmann, Boris A. Malomed, Tomasz Sowiński, and Jakub Zakrzewski 1 Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, Reymonta 4, 30-059 Kraków, Poland 2 Institute of Physics of the Polish Academy of Sciences, Al. Lotników 32/46, PL-02-668 Warsaw, Poland 3 Center for Theoretical Physics of the Pol...

Second-order mappings obtained as reductions of integrable lattice equations are generally expected to have integrals that are ratios of biquadratic polynomials, i.e., to be of QRT-type. In this paper we find reductions of integrable lattice equations that are not of this type. The mappings we consider are exact reductions of integrable lattice equations proposed by Adler, Bobenko and Suris. Su...

We produce a family of new, non-arithmetic lattices in PU(2, 1). All previously known examples were commensurable with lattices constructed by Picard, Mostow, and Deligne– Mostow, and fell into 9 commensurability classes. Our groups produce 5 new distinct commensurability classes. Most of the techniques are completely general, and provide efficient geometric and computational tools for construc...

We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice. We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invaria...

The categorical dualities presented are: (first) for the category of bi-algebraic lattices that belong to variety generated by smallest non-modular lattice with complete (0,1)-lattice homomorphisms as morphisms, and (second) non-trivial (0,1)-lattices belonging same morphisms. Although two categories coincide on their finite objects, essentially differ mostly but not only fact duality second us...