Monadic Boolean Algebras Research Articles

Another look on tense and related operators

Motivated by the classical work of Halmos on functional monadic Boolean algebras, we derive three basic sup-semilattice constructions, among other things, the so-called powersets and powerset operators. Such constructions are extremely useful and can be found in almost all branches of modern mathematics, including algebra, logic, and topology. Our three constructions give rise to four covariant and two contravariant functors and constitute three adjoint situations we illustrate in simple examples.

Monadic Boolean algebras with an automorphism and their relation to $${\mathbf{Df}}_{\mathbf{2}}$$-algebras

In this work, we initiate an investigation of the class $${\mathcal {B}}_{T_km}$$ of monadic Boolean algebras endowed with a monadic automorphism of period k. These algebras constitute a generalization of monadic symmetric Boolean algebras. We determine the congruences on these algebras and we characterize the subdirectly irreducible algebras. This last result allows us to prove that $${\mathcal {B}}_{T_km}$$ is a discriminator variety and as a consequence, the principal congruences are characterized. Finally, we explore, in the finite case, the relationship between this class and the class $${\mathbf{Df}}_{\mathbf{2}}$$ of diagonal-free two-dimensional cylindric algebras.

On Monadic Operators on Modal Pseudocomplemented De Morgan Algebras and Tetravalent Modal Algebras

In our paper, monadic modal pseudocomplemented De Morgan algebras (or mmpM) are considered following Halmos’ studies on monadic Boolean algebras. Hence, their topological representation theory (Halmos–Priestley’s duality) is used successfully. Lattice congruences of an mmpM is characterized and the variety of mmpMs is proven semisimple via topological representation. Furthermore and among other things, the poset of principal congruences is investigated and proven to be a Boolean algebra; therefore, every principal congruence is a Boolean congruence. All these conclusions contrast sharply with known results for monadic De Morgan algebras. Finally, we show that the above results for mmpM are verified for monadic tetravalent modal algebras.

Monadic Algebras: a Standpoint on Rough Sets

The paper studies complete and incomplete information systems, i.e. basic structures of rough set theory (RST), from the standpoint of monadic Boolean algebras (MBAs). In the first part we recall a simple characterisation of RST as a theory of MBAs. In the second part the paper discusses approximation operators obtained from incomplete information systems. In this case we lose a direct link to MBAs and thus our aim is to obtain an MBA which will additionally be interesting from the perspective of RST. In the third part we consider a simple application of approximation operators based on this MBA.

Endomorphisms of monadic Boolean algebras

A classical result about Boolean algebras independently proved by Magill [10], Maxson [11], and Schein [17] says that non-trivial Boolean algebras are isomorphic whenever their endomorphism monoids are isomorphic. The main point of this note is to show that the finite part of this classical result is true within monadic Boolean algebras. By contrast, there exists a proper class of non-isomorphic (necessarily) infinite monadic Boolean algebras the endomorphism monoid of each of which has only one element (namely, the identity), this being the first known example of a variety that is not universal (in the sense of Hedrlin and Pultr), but contains a proper class of non-isomorphic rigid algebras (that is, the identity is the only endomorphism).

Free Modal Lattices via Priestley Duality

A Priestley duality is developed for the variety j ω of all modal lattices. This is achieved by restricting to j ω a known Priestley duality for the variety of all bounded distributive lattices with a meet-homomorphism. The variety j ω was first studied by R. Beazer in 1986. The dual spaces of free modal lattices are constructed, paralleling P.R. Halmos' construction of the dual spaces of free monadic Boolean algebras and its generalization, by R. Cignoli, to distributive lattices with a quantifier.

Free monadic Tarski algebras

A. Monteiro and L. Iturrioz [7], introduced the notion of monadic Tarski algebras as a generalization of the monadic Boolean algebras considered by P. Halmos [6]. They represent the algebraic version of a fragment of the functional monadic calculus. In this paper we determine the structure of the monadic Tarski algebras with a finite set of free generators.

Free Q-distributive lattices

The dual spaces of the free distributive lattices with a quantifier are constructed, generalizing Halmos' construction of the dual spaces of free monadic Boolean algebras.

Construction of monadic three-valued ?ukasiewicz algebras

The notion of monadic three-valued Łukasiewicz algebras was introduced by L. Monteiro ([12], [14]) as a generalization of monadic Boolean algebras. A. Monteiro ([9], [10]) and later L. Monteiro and L. Gonzalez Coppola [17] obtained a method for the construction of a three-valued Łukasiewicz algebra from a monadic Boolea algebra. In this note we give the construction of a monadic three-valued Łukasiewicz algebra from a Boolean algebra B where we have defined two quantification operations ∃ and ∃* such that ∃∀*x=∀*∃x (where ∀*x=-∃*-x). In this case we shall say that ∃ and ∃* commutes. If B is finite and ∃ is an existential quantifier over B, we shall show how to obtain all the existential quantifiers ∃* which commute with ∃. Taking into account R. Mayet [3] we also construct a monadic three-valued Łukasiewicz algebra from a monadic Boolean algebra B and a monadic ideal I of B. The most essential results of the present paper will be submitted to the XXXIX Annual Meeting of the Union Matematica Argentina (October 1989, Rosario, Argentina).

Paul R. Halmos. The basic concepts of algebraic logic. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 9–33. - Paul R. Halmos. Algebraic logic, I. Monadic Boolean algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 37–72. - Paul R. Halmos. Algebraic logic, II. Homogeneous locally finite polyadic Boolean algebras

Paul R. Halmos. The basic concepts of algebraic logic. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 9–33. - Paul R. Halmos. Algebraic logic, I. Monadic Boolean algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 37–72. - Paul R. Halmos. Algebraic logic, II. Homogeneous locally finite polyadic Boolean algebras of infinite degree. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 97–166. - Paul R. Halmos. Algebraic logic, III. Predicates, terms, and operations in polyadic algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 169–209. - Paul R. Halmos. Algebraic logic, IV. Equality in polyadic algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 213–239. - Paul R. Halmos. Polyadic Boolean algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 243–248. - Paul R. Halmos. Predicates, terms, operations, and equality in polyadic Boolean algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 251–257. - Volume 27 Issue 4

Paul R. Halmos. The representation of monadic Boolean algebras. Duke mathematical journal, vol. 26 (1959), pp. 447–454. Reprinted in Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 75–82.

Paul R. Halmos. The representation of monadic Boolean algebras. Duke mathematical journal, vol. 26 (1959), pp. 447–454. Reprinted in <b><i>Algebraic logic</i></b>, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 75–82.

The representation of monadic Boolean algebras

The representation of monadic Boolean algebras