Bjarni Jónsson’s contributions in algebra: an addendum

Normal categories are pointed categorical counterparts of 0-regular varieties, i.e., varieties where each congruence is uniquely determined by the equivalence class of a fixed constant 0. In this paper, we give a new axiomatic approach to normal categories, which uses self-dual axioms on a functor defined using subobjects of objects in the category. We also show that a similar approach can be developed for 0-regular varieties, if we replace subobjects with subsets of algebras containing 0.     

Jónsson posets and unary Jónsson algebras

We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than ${|P|}$ and ${\kappa}$ is a cardinal number strictly less than ${|P|}$ , then P has a principal order ideal of cardinality at least ${\kappa}$ . We apply this result to characterize the possible sizes of unary Jónsson algebras.     

Jónsson posets

According to Kearnes and Oman (2013), a partially ordered set P is Jónsson if it is infinite and the cardinality of every proper initial segment of P is strictly less than the cardinaliy of P. We examine the structure of Jónsson posets.     

A review of some of Bjarni Jónsson’s results on representation of arguesian lattices

We review (and slightly extend) Bjarni Jónsson’s results on representations of arguesian lattices that are complemented, of low height, or of simple gluing structure.     

On successors of Jónsson cardinals

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming does not exist. Namely, if is a Jónsson cardinal then , provided that there is no non-trivial elementary embedding . There are a number of related results in ZFC concerning in V and inner models, for a Jónsson or singular cardinal. Received: 8 December 1998     

Monotone Jónsson operations and near unanimity functions

We define nonextendible colored posets and zigzags of a poset. These notions are related to the earlier notions of gaps, holes, obstructions and zigzags considered by Duffus, Nevermann, Rival, Tardos and Wille. We establish some properties of zigzags. By using these properties we give a proof of the well known conjecture that states that any finite bounded poset which admits Jsson operations, also admits a near unanimity function. We also provide an infinite poset that shows that we cannot drop the finiteness in this conjecture.Presented by R. McKenzie.     

Separability and decidability results for varieties of Jónsson dynamic algebras

Dynamic algebras are algebraic counterparts of dynamic logics: propositional logical systems endowed with a set of modal operators. In [18], B. Jónsson introduced dynamic algebras as Boolean algebras with unary operators, the indices of which range over a given Kleene algebra. On the other hand, V.R. Pratt and D. Kozen proposed a two-sorted approach to dynamic algebras, which was followed in the early papers on the topic, such as Fischer and Ladner [15] and Németi [28]. For a recent overview of the field cf. [4]. In the present paper we investigate connections (as well as diversities) between these two approaches. Our main aim is to transfer (where possible) two-sorted results on separability and decidability to the one-sorted case and to extend them to broad classes of varieties of Jónsson dynamic algebras. In particular, as a consequence of such considerations, we obtain a decidability result on Kleene algebras.     

Generators for complemented modular lattices and the von Neumann–Jónsson coordinatization theorems

Extending work of von Neumann, Jónsson has shown that each complemented modular lattice, L admitting a large partial n-frame with n ≥ 4, or with n ≥ 3 and L Arguesian, can be coordinatized as the lattice of all principal right ideals of some regular ring. His proof built on the embedding of L into the subgroup lattice of an abelian group which follows from Frink’s embedding of L into to a direct product of subspace lattices of irreducible projective spaces and coordinatization of the latter. We offer a proof which, in addition to these results, employs only some elementary linear algebra. Luca Giudici’s thesis [6] is an important source for this approach.     

Jónsson's theorem in non-modular varieties

A version of Jónsson's Theorem, as previously generalized, holds in non-modular varieties. Received March 20, 2000; accepted in final form December 20, 2000.     

A Geometric Approach to the Calderón-Zygmund Estimates

In this note, we will show a new proof of the classical Calderón-Zygmund estimates established in [7] 1952. The estimates are among the fundamental estimates for elliptic equations. Parabolic equations are also considered.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and Lipschitz function b∈Λβ0(Rn) is discussed from Lp(Rn) to Lq(Rn), 1/q=1/p-β0/n, and from Lp(Rn) to Triebel-Lizorkin space Fβ0,∞p. We also obtain the boundedness of generalized Toeplitz operatorθbα0 from LP(Rn) to Lq(Rn), 1/q =1/p-α0 β0/n. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and BMO function b is discussed on LP(Rn), 1 < p <∞.     

Be n Fixed,or go to Infinity: Difficulties are Welcome,and Almost Surely Overcome; A Tribute to Endre Csáki

Periodica Mathematica Hungarica -     

Complex J—Symplextic Geometry With Application to Ordinary Differential Operators

In this paper, we deal with complex J-symplectic geometry with application to ordinarydifferential operators. We define complex J-symplectic spaces and their J-Lagrangian subspacesand complete J-Lagrangian subspaces, and then we discuss their basic algebraic properties. Thenwe apply them to the theory of J-selfadjoint operators and give J-symplectic geometry completecharacterizations of J-selfadjoint extensions of J-symmetric operatorsDefinition 1 A complex J-symplectic space S is a compl…