Lattice-ordered matrix algebras containing positive cycles

It is shown that if a lattice-ordered n × n (n ≥ 2) matrix ring over a totally ordered integral domain or division ring containing a positive n-cycle, then it is isomorphic to the lattice-ordered n × n matrix ring with entrywise lattice order.     

Lattice orders on 2 × 2 triangular matrix algebras

We construct all the lattice orders on a 2 × 2 triangular matrix algebra over a totally ordered field that make it into a lattice-ordered algebra. It is shown that every lattice order in which the identity matrix is not positive may be obtained from a lattice order in which the identity matrix is positive.     

Finite-dimensional <Emphasis Type="Italic">ℓ</Emphasis>-simple lattice-ordered algebras with a <Emphasis Type="Italic">d</Emphasis>-basis

It is shown that a unital finite-dimensional ℓ-simple ℓ-algebra with a distributive basis is isomorphic to a lattice-ordered matrix algebra with the entrywise lattice order over a lattice-ordered twisted group algebra of a finite group with the coordinatewise lattice order. It is also shown that the isomorphism is unique.     

Construction of lattice orders on the semigroup ring of a positive rooted monoid

Lattice orders on the semigroup ring of a positive rooted monoid are constructed, and it is shown how to make the monoid ring into a lattice-ordered ring with squares positive in various ways. It is proved that under certain conditions these are all of the lattice orders that make the monoid ring into a lattice-ordered ring. In particular, all of the partial orders on the polynomial ring A[x] in one positive variable are determined for which the ring is not totally ordered but is a lattice-ordered ring with the property that the square of every element is positive. In the last section some basic properties of d-elements are considered, and they are used to characterize lattice-ordered division rings that are quadratic extensions of totally ordered division rings.     

Generalized Boolean Algebras in Lattice-Ordered Groups

In this paper, characterizations are given for the free lattice-ordered group over a generalized Boolean algebra and the freel -module of a totally ordered integral domain with unit over a generalized Boolean algebra. Extensions of lattice-ordered groups using generalized Boolean algebras are defined and their properties studied.     

Positive derivations on archimedean lattice-ordered rings

It is known that the only positive derivation on a reduced archimedean f-ring is the zero derivation. We investigate derivations on general archimedean lattice-ordered rings. First, we consider semigroup rings over cyclic semigroups and show that, in the finite case, the only derivation that is zero on the underlying ring is the zero derivation and that, in the infinite case, such derivations are always based on the derivative. Turning our attention to lattice-ordered rings, we show that, on many algebraic extensions of totally ordered rings, the only positive derivation is the zero derivation and that, for transcendental extensions, derivations that are lattice homomorphisms are always translations of the usual derivative and derivations that are orthomorphisms are always dilations of the usual derivative. We also show that the only positive derivation on a lattice-ordered matrix ring over a subfield of the real numbers is the zero derivation, and we prove a similar result for certain lattice-ordered rings with positive squares. The second author thanks Hamilton College for its support of his visits to the first author in Houston. He also thanks John Miller for his friendship and hospitality over the last thirty years.     

Lattice-theoretic properties of algebras of logic

In the theory of lattice-ordered groups, there are interesting examples of properties — such as projectability — that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semi-linear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a,1]$[a,1]$ is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety.     

Lattice-Ordered Matrix Rings Over the Integers

We show that the only compatible lattice order on a matrix ring over the integers for which the identity matrix is positive is (up to isomorphism) the usual, entrywise, lattice order. We also find a condition that guarantees that the only compatible lattice order on a matrix ring over the integers is formed by multiplying the positive cone of the usual, entrywise, lattice order by a matrix with positive entries. Using this condition, we show that such orders are the only compatible ones in the two-by-two case.     

Division Closed Lattice-Ordered Rings

In this note we consider properties of unital lattice-ordered rings that are division closed and characterize unital lattice-ordered algebras that are algebraic and division closed. Extending partial orders to lattice orders that are division closed is also studied. In particular, it is shown that a field is L ^? if and only if it is O ^?.     

On Subrings of Simple Artinian Rings

For an n × n matrix ring over a division ring we investigate the structure of all subrings which have the property that every projection onto a component is either the zero map or surjective. It turns out that many of these rings are semiprimary.     

Prime Radicals of Lattice κ-Ordered Algebras

The Kopytov order for any algebra over a field is considered. Necessary and sufficient conditions for an algebra to be a linearly ordered algebra are presented. Some results concerning the properties of ideals of linearly ordered algebras are obtained. Some examples of algebras with the Kopytov order are described. The Kopytov order for these examples induces the order on other algebraic objects. The purpose of this paper is to investigate a generalization of the concept of prime radical to lattice-ordered algebras over partially ordered fields. Prime radicals of l-algebras over partially ordered and directed fields are described. Some results concerning the properties of the lower weakly solvable l-radical of l-algebras are obtained. Necessary and sufficient conditions for the l-prime radical of an l-algebra to be equal to the lower weakly solvable l-radical of the l-algebra are presented.     

Value Functions on Simple Artinian Rings

In this paper we introduce the notion of a value function ω on a simple Artinian ring Q whose restriction to the center of Q is a Krull valuation. We then study the structure of a valuation ring Bω corresponding to ω and its ideals. It is shown that Bω is a prime Goldie ring and the set of two-sided ideals of Bω is totally ordered by inclusion with the maximal ideal J(BW). Furthermore, it is proved that if Q is a central simple algebra, then the valuation ring Bω is a Dubrovin valuation ring which is integral over its center.     

On inverse spectrum problems for normal nonnegative matrices

For a set σ with n complex numbers, some sufficient conditions are found for σ to be the spectrum of an n ×n normal (entrywise) nonnegative (positive) matrix. After proving a fundamental theorem and introducing the companion set σ′ of σ which consists of real numbers, we prove that if σ′ satisfies any known sufficient conditions for a real set to be the spectrum of a nonnegative matrix introduced by Suleimanova, Perfect, Salzmann and Kellogg respectively, then σ is the spectrum of an n×n normal nonnegative matrix.     

Recognition of Lattice-Ordered Matrix Rings

For an ?-unital ?-ring R, various recognition criteria are given for R to be isomorphic to a matrix ?-ring over an ?-unital ?-ring with the entrywise order.     

On Strongly J-Clean Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. A ring is strongly J-clean in case each of its elements is strongly J-clean. We investigate, in this article, strongly J-clean rings and ultimately deduce strong J-cleanness of T _n (R) for a large class of local rings R. Further, we prove that the ring of all 2 × 2 matrices over commutative local rings is not strongly J-clean. For local rings, we get criteria on strong J-cleanness of 2 × 2 matrices in terms of similarity of matrices. The strong J-cleanness of a 2 × 2 matrix over commutative local rings is completely characterized by means of a quadratic equation.     

Lattice-Ordered Matrix Algebras Over Real GCD-Domains

Let R ? ? be a GCD-domain. In this article, Weinberg's conjecture on the n × n matrix algebra M _n (R) (n ≥ 2) is proved. Moreover, all the lattice orders (up to isomorphisms) on a full 2 × 2 matrix algebra over R are obtained.     

Lattice-ordered fields

Mainly archimedean lattice-ordered fields (l-fields) are investigated in this paper. An archimedean l-field has a largest subfield (its o-subfield) which can be totally ordered in such a way that the l-field is a partially ordered vector space over this subfield. For archimedean l-fields which are algebraic over their o-subfields the following questions are investigated: What is the structure of the additive l-group of an l-field? Can the lattice order of an l-field be extended to a total order? Are the intermediate fields of an l-field and its o-subfield also l-fields with the induced partial order?     

Theory of cones

This survey deals with the aspects of archimedian partially ordered finite-dimensional real vector spaces and order preserving linear maps which do not involve spectral theory. The first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations. The study of inequalities resulted in the definition of a polyhedral cone K and its face lattice F(K). In Section II.A the face lattice of a not necessarily polyhedral cone K in a vector space V is investigated. In particular the interplay between the lattice properties of $\text{F}$(K) and geometric properties of K is emphasized. Section II.B turns to the cones Π(K) in the space of linear maps on V. Recall that Π(K) is the cone of all order preserving linear maps. Of particular interest are the algebraic structure of Π(K) as a semiring and the nature of the group Aut(K) of nonsingular elements A?Π(K) for which A^-1?Π(K) as well. In a short final section the cone P_n of n×n positive semidefinite matrices is discussed. A characterization of the set of completely positive linear maps is stated. The proofs will appear in a forthcoming paper.     

On Rings Whose Reflexive Ideals are Principal

We call a prime Noetherian maximal order R a pseudo-principal ring if every reflexive ideal of R is principal. This class of rings is a broad class properly containing both prime Noetherian pri-(pli) rings and Noetherian unique factorization rings (UFRs). We show that the class of pseudo-principal rings is closed under formation of n × n full matrix rings. Moreover, we prove that if R is a pseudo-principal ring, then the polynomial ring R[x] is also a pseudo-principal ring. We provide examples to illustrate our results.     

Nil-Armendariz Rings Relative to a Monoid

For a monoid M, we introduce nil-Armendariz rings relative to M, which are a generalization of nil-Armendariz and M-Armendariz rings, and investigate their properties. First we show that semicommutative rings are nil-Armendariz relative to every unique product monoid M. Also it is shown that for a strictly totally ordered monoid M and an ideal I of R, if I is a semicommutative subrng of R and R/I nil-Armendariz relative to M, then R is nil-Armendariz relative to M. Then we show that if R is a semicommutative ring and nil-Armendariz relative to M, then R is nil-Armendariz relative to M × N, where N is a unique product monoid. As corollaries we obtain some results of [2] and [10].