Small orthomodular partial algebras

Orthomodular partial algebras (OMAs) can be seen as the algebraic representation of orthomodular posets. We use Greechie diagrams for the graphical representation of OMAs and investigate characterizations for the strong embeddability of a given OMA into a Boolean OMA. We present a complete list of the Greechie diagrams of OMAs up to 24 elements, and we show that there exists an infinite OMA that is generated by 4 elements.     

Boolean deductive systems of BL-algebras

BL-algebras rise as Lindenbaum algebras from many valued logic introduced by Hájek [2]. In this paper Boolean ds and implicative ds of BL-algebras are defined and studied. The following is proved to be equivalent: (i) a ds D is implicative, (ii) D is Boolean, (iii) L/D is a Boolean algebra. Moreover, a BL-algebra L contains a proper Boolean ds iff L is bipartite. Local BL-algebras, too, are characterized. These results generalize some theorems presented in [4], [5], [6] for MV-algebras which are BL-algebras fulfiling an additional double negation law x = x ^**. Received: 22 June 1998 /?Published online: 18 May 2001     

Subalgebras of Orthomodular Lattices

Sachs (Can J Math 14:451–460, 1962) showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham (Int J Theor Phys 37(11):2669–2733, 1998, Int J Theor Phys 38(3):827–859, 1999), at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to this line of work.     

Order topology in vector lattices with closure by one step

An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step” that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets of the interval (0, 1)) is given. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15 1995, pp. 213–220.     

An example of a commutative basic algebra which is not an MV-algebra

Many algebras arising in logic have a lattice structure with intervals being equipped with antitone involutions. It has been proved in [CHK1] that these lattices are in a one-to-one correspondence with so-called basic algebras. In the recent papers [BOTUR, M.—HALAŠ, R.: Finite commutative basic algebras are MV-algebras, J. Mult.-Valued Logic Soft Comput. (To appear)]. and [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] we have proved that every finite commutative basic algebra is an MV-algebra, and that every complete commutative basic algebra is a subdirect product of chains. The paper solves in negative the open question posed in [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] whether every commutative basic algebra on the interval [0, 1] of the reals has to be an MV-algebra.     

Some Boolean Algebras with Finitely Many Distinguished Ideals I

We consider the theory Th_prin of Boolean algebras with a principal ideal, the theory Th_max of Boolean algebras with a maximal ideal, the theory Th_ac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Th_sa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Th_prin and Th_sa. If T is a theory in a first order language and α is a linear order with least element, then we let Sentalg(T) be the Lindenbaum-Tarski algebra with respect to T, and we let intalg(α) be the interval algebra of α. Using rank diagrams, we show that Sentalg(Th_prin) ? intalg(ω⁴), Sentalg(Th_max) ? intalg(ω³) ? Sentalg(Th_ac), and Sentalg(Th_sa) ? intalg(ω² + ω²). For Th_max and Th_ac we use Ershov's elementary invariants of these theories. We also show that the algebra of formulas of the theory Tx of Boolean algebras with finitely many ideals is atomic.     

Non-embeddable simple relation algebras

This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation algebra an element x satisfying the condition (a) must be an atom of . It follows that x must also be an atom in every simple extension of . Andréka, Jónsson and Németi [1, Problem 4] (see [12, Problem P5]) asked whether the converse holds: if x is an atom in every simple extension of a simple relation algebra, must it satisfy (a)? We show that the answer is “no”.? The only known examples of simple relation algebras without simple proper extensions are the algebras of all binary relations on a finite set. Jónsson proposed finding all finite simple relation algebras without simple proper extensions [12, Problem P6]. We show how to construct many new examples of finite simple relation algebras that have no simple proper extensions, thus providing a partial answer for this second problem. These algebras are also integral and nonrepresentable.? Andréka, Jónsson, Németi [1, Problem 2] (see [12, Problem P7]) asked whether there is a countable simple relation algebra that cannot be embedded in a one-generated relation algebra. The answer is “yes”. Givant [3, Problem 9] asked whether there is some k such that every finitely generated simple relation algebra can be embedded in a k-generated simple relation algebra. The answer is “no”. Received November 27, 1996; accepted in final form July 3, 1997.     

Hyper‐Archimedean BL‐algebras are MV‐algebras

Generalizations of Boolean elements of a BL‐algebra L are studied. By utilizing the MV‐center MV(L) of L, it is reproved that an element x ∈ L is Boolean iff x ∨ x * = 1 . L is called semi‐Boolean if for all x ∈ L, x * is Boolean. An MV‐algebra L is semi‐Boolean iff L is a Boolean algebra. A BL‐algebra L is semi‐Boolean iff L is an SBL‐algebra. A BL‐algebra L is called hyper‐Archimedean if for all x ∈ L, xⁿ is Boolean for some finite n ≥ 1. It is proved that hyper‐Archimedean BL‐algebras are MV‐algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV‐algebras or BL‐algebras. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Volterra-like Banach Algebras and their Second Duals

 This paper presents and studies a class of algebras which includes the usual Volterra algebra. Roughly speaking, they relate to the Volterra algebra in the way a general locally compact group relates to ℝ. We show that they can be viewed as quotients of some semigroup algebras introduced by Baker and Baker [1]. Their sets of nilpotent elements are dense. We investigate the second duals of these algebras and find that most of the properties found in [7] for the biduals of the group algebras L ¹(G) for compact G are retained here. Received 8 July 1997; in revised form 17 November 1997     

Condition for inducing topology in a vector lattice by the order completion

Properties of an order topology in vector lattices and Boolean algebras are studied. The main result is the following: in a vector lattice or a Boolean algebra with the condition of “closure by one step” (a generalization of the well-known “regularity” property of Boolean algebras and K-spaces) the order topology is induced by the topology of its Dedekind completion. Bibliography: 4 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16, 1997, pp. 204–207.     

Boundaries of weak peak points in noncommutative algebras of Lipschitz functions

It has been shown that any Banach algebra satisfying ‖f ²‖ = ‖f‖² has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, \mathbbF\mathbb{F}) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where \mathbbF\mathbb{F} = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, \mathbbF\mathbb{F}) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.     

Complexity of Boolean algebras and their Scott rank

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras. Supported by RFFR grant No. 99-01-00485, by a grant for Young Scientists from SO RAN, 1997, and by the Federal Research Program (FRP) “Integration”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 643–666, November–December, 1999.     

Coverings of [MO n ] and minimal orthomodular lattices

If T is an orthomodular lattice (OML), we denote by [T] the equational class generated by T. In this paper we characterize the finite OMLs T such that [T] covers some [MO _n ]. These OMLs T are the non-modular OMLs such that all proper sub-OMLs of T are modular. An OML satisfying that last property is called minimal. There exist infinitely many minimal OMLs provided by quadratic spaces over finite fields. We describe them and give a new way to represent their Greechie diagrams in two separate parts. Other methods to obtain finite minimal OMLs are given. Received May 14, 2005; accepted in final form May 30, 2007.     

Division algebras with a projective basis

Letk be any field andG a finite group. Given a cohomology class α∈H ²(G,k ^*), whereG acts trivially onk ^*, one constructs the twisted group algebrak ^αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z _n×Z_n). This paper has two main results: First we prove that ifD=k ^α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k ^α G is a product of cyclic algebras. Furthermore, ifD _p is ap-primary factor ofD, thenD _p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD _p, p odd. Ifp=2 we show thatD ₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z ₂×Z₂n.     

The spectrum of partitions of a Boolean algebra

The main notion dealt with in this article is

Terms that are self-dual in Boolean algebras but not in MO 2

An absorption law is an identity of the form p = x. The ternary function x+y+z (ring addition) in Boolean algebras satisfies three absorption laws in two variables. If a term satisfies these three identities in a variety, it is called a minority term for that variety. We construct a minority term p for orthomodular lattices such the identity defines Boolean algebras modulo orthomodular lattices. (The dual of p is denoted by .) Consequently, having a unique minority term function characterizes Boolean algebras among orthomodular lattices. Our main result generalizes this example to arbitrary arity and arbitrary consistent sets of 2-variable absorption laws. Presented by J. Berman.     

The preservation of Sahlqvist equations in completions of Boolean algebras with operators

Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators. Under the assumption that the operators of such an algebra are completely additive, he showed that the completion of always exists and is unique up to isomorphisms over . Moreover, strictly positive equations are preserved under completions a strictly positive equation that holds in must hold in the completion of . In this paper we extend Monk’s preservation theorem by proving that certain kinds of Sahlqvist equations (as well as some other types of equations and implications) are preserved under completions. An example is given that shows that arbitrary Sahlqvist equations need not be preserved. Received May 3, 1998; accepted in final form October 7, 1998.     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subalgebra extensions of partial monounary algebras

For a subalgebra B of a partial monounary algebra A we define the quotient partial monounary algebra A/B. Let B, B, C be partial monounary algebras. In this paper we give a construction of all partial monounary algebras A such that B is a subalgebra of A and C ≅ A/B.