Regular double p-algebras

In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDPn, n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP1, LV(RDP1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP1 and conclude that LV(RDP1) is countably infinite. An equational basis for each proper subvariety of RDP1 is given. To study the subvarieties RDPn with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2ℵ0. We also use Priestley duality to prove that RDP and each of its subvarieties RDPn are generated by their finite members.

THERE IS NO FINITE-VARIABLE EQUATIONAL AXIOMATIZATION OF REPRESENTABLE RELATION ALGEBRAS OVER WEAKLY REPRESENTABLE RELATION ALGEBRAS

We prove that any equational basis that defines representable relation algebras (RRA) over weakly representable relation algebras (wRRA) must contain infinitely many variables. The proof uses a construction of arbitrarily large finite weakly representable but not representable relation algebras whose “small” subalgebras are representable.

Equations in Type-2 Fuzzy Sets

The main concern of this paper is with the equations satisfied by the algebra of truth values of type-2 fuzzy sets. That algebra has elements all mappings from the unit interval into itself with operations given by certain convolutions of operations on the unit interval. There are a number of positive results. Among them is a decision procedure, similar to the method of truth tables, to determine when an equation holds in this algebra. One particular equation that holds in this algebra implies that every subalgebra of it that is a lattice is a distributive lattice. It is also shown that this algebra is locally finite. Many questions are left unanswered. For example, we do not know whether or not this algebra has a finite equational basis, that is, whether or not there is a finite set of equations from which all equations satisfied by this algebra follow. This and various other topics about the equations satisfied by this algebra will be discussed.

EQUATIONAL BASES FOR SOME 0-DIRECT UNIONS OF SEMIG OUPS

In this paper,we investigate identities satis .ed by 0-direct unions of a semigroup with its anti-isomorphic copy,which serve as the standard tool for showing that an arbitrary semigroup can be embedded in (a semigroup reduct of)an involution semigroup.We show that,given the set of semigroup identities they satisfy,the involution de .ned on such 0-direct unions can be captured by only two additional identities involving the unary operation symbol.As a corollary of a result on .niteness of equational bases for such involution semigroups,we present an involution semigroup (which is,however,not an inverse one)consisting of 13 elements and not having a .nite equational basis.

The Max-Plus Algebra of the Natural Numbers has no Finite Equational Basis

This paper shows that the collection of identities which hold in<br />the algebra N of the natural numbers with constant zero, and binary<br />operations of sum and maximum is not finitely based. Moreover, it<br />is proven that, for every n, the equations in at most n variables that<br />hold in N do not form an equational basis. As a stepping stone in<br />the proof of these facts, several results of independent interest are<br />obtained. In particular, explicit descriptions of the free algebras in the<br />variety generated by N are offered. Such descriptions are based upon<br />a geometric characterization of the equations that hold in N, which<br />also yields that the equational theory of N is decidable in exponential<br />time.

A Self-Dual Equational Basis for Boolean Algebras

The principle of duality for Boolean algebra states that if an identity ƒ = g is valid in every Boolean algebra and if we transform ƒ = g into a new identity by interchanging (i) the two lattice operations and (ii) the two lattice bound elements 0 and 1, then the resulting identity ƒ = g is also valid in every Boolean algebra. Also, the equational theory of Boolean algebras is finitely based. Believing in the cosmic order of mathematics, it is only natural to ask whether the equational theory of Boolean algebras can be generated by a finite irredundant set of identities which is already closed for the duality mapping. Here we provide one such equational basis.

Varieties of Orthomodular Lattices

In this paper we start investigating the lattice of varieties of orthomodular lattices. The varieties studied here are those generated by orthomodular lattices which are the horizontal sum of Boolean algebras. It turns out that these form a principal ideal in the lattice of all varieties of orthomodular lattices. We give a complete description of this ideal; in particular, we show that each variety in it is generated by its finite members. We furthermore show that each of these varieties is finitely based by exhibiting a (rather complicated) finite equational basis for each variety.Our methods rely heavily on B. Jonsson's fundamental results in [8]. This, however, could be avoided by starting out with the equations given in sections 3 and 4. Some of our arguments were suggested by Baker [1],