Subdirect Product Research Articles

The Intermediate Value Theorem for Polynomials over Lattice‐ordered Rings of Functions

ABSTRACTThe classical intermediate value theorem for polynomials with real coefficients is generalized to the case of polynomials with coefficients in a lattice‐ordered ring that is a subdirect product of totally ordered rings. Several candidates for a generalization are investigated, and particular attention is paid to the case when the lattice‐ordered ring is the algebra C(X) of continuous real‐valued functions on a completely regular topological space X. For all but one of these generalizations, the intermediate value theorem holds only if X is an F‐space in the sense of Gillman and Jerison. Surprisingly, for the most interesting of these generalizations, if X is compact, the intermediate value theorem holds only if X is an F‐space and each component of X is an hereditarily indecomposable continuum. It is not known if there is an infinite compact connected space in which this version of the intermediate value theorem holds.

Subdirect products of finite groups with Hall π-subgroups

Subdirect products of finite groups with Hall π-subgroups

Birkhoff-like sheaf representation for varieties of lattice expansions

Given a variety ν we study the existence of a class ℱ such that S1 every A e ν can be represented as a global subdirect product with factors in ℱ and S2 every non-trivial A e ℱ is globally indecomposable. We show that the following varieties (and its subvarieties) have a class ℱ satisfying properties S1 and S2: p-algebras, distributive double p-algebras of a finite range, semisimple varieties of lattice expansions such that the simple members form a universal class (bounded distributive lattices, De Morgan algebras, etc) and arithmetical varieties in which the finitely subdirectly irreducible algebras form a universal class (f-rings, vector groups, Wajsberg algebras, discriminator varieties, Heyting algebras, etc). As an application we obtain results analogous to that of Nachbin saying that if every chain of prime filters of a bounded distributive lattice has at most length 1, then the lattice is Boolean.

Large subdirect products of flat modules

(1996). Large subdirect products of flat modules. Communications in Algebra: Vol. 24, No. 4, pp. 1389-1407.

Locally Boolean spectra

In [35] we study three types of representability of algebras by sheaves: the global representability, the Stone representability and the locally Boolean representability. In that paper the study is centered in the two first of the three notions cited above, and the characterizations are given in terms of properties of the congruence lattice of the algebra. This paper is a continuation of [35] in which we study with the same approach the third of the above three notions: the locally Boolean representability. The central concepts of [35] are those of compactness of a set of congruences, projectives and (appropriate versions of) the Chinese Remainder Theorem. An unexpected fact is that the first two of the three concepts above are precisely the central concepts in the study of the locally Boolean representability. The generality of the results obtained in both works puts in perspective the power of the universal algebra in the study of sheaf representations of algebras. As it can be noted in the applications, in all cases the results are obtained by combining a minute amount of particular algebra with strong general universal algebra theorems. In Section 1 we fix notation and terminology and review a few basic facts which will be needed later. In Section 2 we define the notion of compactness of a set of congruences and prove a key result of the paper (Lemma 2.1). In Section 3 we define locally Boolean subdirect products and show the relation between this concept and sheaves. In Section 4 we develop, in a similar way as in Section 4 of [35], the general construction presented in [26], and we obtain some results on compactness of a set of congruences. In Section 5 we associate with each universal class de' a discriminator variety called the Jg-spectral variety which is such that the locally Boolean representations with /'actors in X-{trivial algebras} are just the irredundant fully expanded subdirect representations by simple algebras in the tit-spectral variety (Theorem 5.3). As an application we obtain that if ~ ' is a V3 class, then the class of locally Boolean d/Z-representable algebras is closed under ultraproducts and direct products. Finally, we use a result of R. McKenzie to

A metatheorem for deriving fuzzy theorems from crisp versions

A metatheorem is developed that allows theorems containing references to crisp substructures to be reformulated to refer to fuzzy substructures. Theorems concerning fuzzy substructures in semigroups, rings, and lattices are derived in concise fashion from analogous theorems that concern crisp substructures. The modularity of the lattice of fuzzy normal subgroups of a group is concisely derived from the modularity of its lattice of crisp normal subgroups. The fundamental tool used is a function that represents the algebra of fuzzy subsets of a set as a subdirect product of copies of the algebra of crisp subsets.

On the Periodic Radical of a Ring

AbstractLet R be a ring and P(R) the sum of all periodic ideals of R. We prove that P(R) is the intersection of all prime ideals Pα such that contains no nontrivial periodic ideals. We also prove that P(R) = 0 if and only if Rs is a subdirect product of prime rings Rα with P(Rα) = 0.

Subdirect Products of Totally Ordered Implicative Algebras

Subdirect Products of Totally Ordered Implicative Algebras

Properties of algebras of primary order with one ternary Mal'tsev operation

We study properties of free solvable p-algebras and of p-algebras of primary orders. It is shown that a free solvable p-algebra is embeddable in a subdirect product of algebras with these orders. An example of a simple non-Abelian p-algebra of a primary order is constructed, and we also give an example of a solvable non-nilpotent p-algebra of any finite order not less than 6.

Groupable lattices

A lattice is called groupable provided it can be endowed with the structure of an l-group (lattice ordered group). The primary objective of this paper is to introduce an order theoretic property of groupable lattices which implies that all associated l-groups are subdirect products of totally ordered groups. This is an analog to Iwasawa's well-known result which asserts that a conditionally complete l-group is abelian. A secondary objective is to outline a general method for identifying classes of l-groups determined by order theoretic properties.

The structure of commutative rings with monomorphic flat envelopes

It is obvious that OF and Von Neumann regular rings have monomorphic flat envelopes. In this paper we completely describe the structure,in terms of OF and Von Neumann regular rings, of those commutative rings all of whose modules have a monomorphic flat envelope (m.f.e. ). For that, we introduce the notion of locally QF ring with m.f.e., whose structure is given in terms of OF rings. It turns out that a commutative ring R with m.f.e. is characterized as a (essential) subdirect product of a locally QF ring with m.f.e. and a Von Neumann regular ring, with the latter flat as an R-module.

Quasi‐median graphs and algebras

AbstractScattered through the literature various structures occur that generalize median graphs and median algebras: quasi‐median graphs, retracts of Hamming graphs, graphs of finite windex, isotropic media, subdirect products of simplex algebras, certain ternary algebras, and so on. We connect them all up, provide some new generalizations of quasi‐median graphs, some new sets of axioms for quasi‐median algebras, and some shorter proofs as well.

Lattice-Ordered Algebras that are Subdirect Products of Valuation Domains

An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if $A/P$ is a valuation domain for every prime ideal P of A . If M is a maximal $\ell$-ideal of A, then the rank of A at M is the number of minimal prime ideals of A contained in M , rank of A is the sup of the ranks of A at each of its maximal $\ell$-ideals. If the latter is a positive integer, then A is said to have finite rank, and if $A = C(X)$ is the ring of all real-valued continuous functions on a Tychonoff space, the rank of X is defined to be the rank of the f-ring $C(X)$, and X is called an SV-space if $C(X)$ is an ST-ring. X has finite rank k iff k is the maximal number of pairwise disjoint cozero sets with a point common to all of their closures. In general f-rings these two concepts are unrelated, but if A is uniformly complete (in particular, if $A = C(X)$) then if A is an SV-ring then it has finite rank. Showing that this latter holds makes use of the theory of finite-valued lattice-ordered (abelian) groups. These two kinds of rings are investigated with an emphasis on the uniformly complete case. Fairly powerful machinery seems to have to be used, and even then, we do not know if there is a compact space X of finite rank that fails to be an SV-space.

The Unitability of l-Prime Lattice-Ordered Rings with Squares Positive

It is shown that an i-prime lattice-ordered ring with squares positive and an f-superunit can be embedded in a unital i-prime lattice-ordered ring with squares positive. In [1, p. 325] Steinberg asked if an i-prime i-ring with squares positive and an f-superunit can be embedded in a unital i-prime i-ring with squares positive. In this paper we show that the answer is yes. If R is a lattice-ordered ring (i-ring) and a E R+, then a is called an felement of R if b A c = 0 implies ab A c = ba A c = 0. Let T = T(R) = {a E R: lal is an f-element of R}. Then T is a convex f-subring of R, and R is a subdirect product of totally ordered T-T bimodules [2, Lemma 1]. R is an f-ring precisely when T = R. Throughout this paper T will denote the subring of f-elements of R. An element e > 0 of an i-ring R is called a superunit if ex > x and xe > x for each x E R+; e is an f-superunit if it is a superunit and an f-element. R is infinitesimal if x2 0 for each a in R. Let A be any ring and a E A,. If a satisfies ab = ba = nb for some fixed integer n and all b E A, then a is said to be an n-fier of A and n is said to have an n-fier a in A. Let K = {n E Z: n has n-fiers in A}. Then K is an ideal in the ring Z of integers. The ideal K is called the modal ideal of A; its nonnegative generator k is called the mode of A. If R is an f-ring with mode k > 0, then R has a unique k-fier x > 0 [3, III, Lemma 2.1]. Let R be an /-ring, and let S = S(R) = {a E R: lal > d, Vd E T+} and T' = {a E R: lal A d = 0, Vd E T+}. It is clear that T' is a convex /-subgroup of R. The referee has pointed out to us that the following lemma, on which our arguments are based, is a special case of [4, Lemma 6.2], and P. Conrad attributes it to A. H. Clifford. Received by the editors July 20, 1992 and, in revised form, October 30, 1992. 1991 Mathematics Subject Classification. Primary 06F25; Secondary 1 6A86.

Cover-preserving embedding of modular lattices

In this note we prove: If a subdirect product of finitely many finite projective geometries has the cover-preserving embedding property, then so does each factor.

The congruence lattice of a combinatorial strict inverse semigroup

The congruence lattice of a combinatorial strict inverse semigroup is shown to be isomorphic to a complete subdirect product of congruence lattices of semilattices preserving pseudocomplements.

A restricted full subdirect product of aloebras

In the present paper we introduce the concept of an L -restrictecd fu11 subdirect product of algebras. This notion is a common generalization of full subdirect and weak direct product. The main result of our paper of a uniqueness theorem for restricted full subdirect representations of algebras. This result implies some theorems on weak direct and full subdirect representations.

Infinite families of isomorphic nonconjugate finitely generated subgroups

Let ⟨ , ⟩ : L × L → Z \langle \;,\;\rangle :L \times L \to \mathbb {Z} be a nondegenerate symmetric bilinear form on a finitely generated free abelian group L which splits as an orthogonal direct sum ( L , ⟨ , ⟩ ) ≅ ( L 1 , ⟨ , ⟩ ) ⊥ ( L 2 , ⟨ , ⟩ ) ⊥ ( L 3 , ⟨ , ⟩ ) (L,\;\langle \;,\;\rangle ) \cong ({L_1},\;\langle \;,\;\rangle ) \bot ({L_2},\;\langle \;,\;\rangle ) \bot ({L_3},\;\langle \;,\;\rangle ) in which ( L 1 , ⟨ , ⟩ ) ({L_1},\;\langle \;,\;\rangle ) has signature (2, 1), ( L 2 , ⟨ , ⟩ ) ({L_2},\;\langle \;,\;\rangle ) has signature (n, 1) with n ≥ 2 n \geq 2 , and ( L 3 , ⟨ , ⟩ ) ({L_3},\;\langle \;,\;\rangle ) is either zero or indefinite with rk Z ( L 3 ) ≥ 3 {\text {rk}}_\mathbb {Z}({L_3}) \geq 3 . We show that the integral automorphism group Aut Z ( L , ⟨ , ⟩ ) {\operatorname {Aut} _\mathbb {Z}}(L,\;\langle \;,\;\rangle ) contains an infinite family of mutually isomorphic finitely generated subgroups ( Γ σ ) σ ∈ Σ {({\Gamma _\sigma })_{\sigma \in \Sigma }} , no two of which are conjugate. In the simplest case, when L 3 = 0 {L_3} = 0 , the groups Γ σ {\Gamma _\sigma } are all normal subdirect products in a product of free groups or surface groups. The result can be seen as a failure of the rigidity property for subgroups of infinite covolume within the corresponding Lie group Aut Z ( L ⊗ Z R , ⟨ , ⟩ ⊗ 1 ) {\operatorname {Aut} _\mathbb {Z}}(L{ \otimes _\mathbb {Z}}\mathbb {R},\;\langle \;,\;\rangle \otimes 1) .

Lattice-ordered algebras that are subdirect products of valuation domains

An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if A / P A/P is a valuation domain for every prime ideal P of A . If M is a maximal ℓ \ell -ideal of A, then the rank of A at M is the number of minimal prime ideals of A contained in M , rank of A is the sup of the ranks of A at each of its maximal ℓ \ell -ideals. If the latter is a positive integer, then A is said to have finite rank, and if A = C ( X ) A = C(X) is the ring of all real-valued continuous functions on a Tychonoff space, the rank of X is defined to be the rank of the f-ring C ( X ) C(X) , and X is called an SV-space if C ( X ) C(X) is an ST-ring. X has finite rank k iff k is the maximal number of pairwise disjoint cozero sets with a point common to all of their closures. In general f-rings these two concepts are unrelated, but if A is uniformly complete (in particular, if A = C ( X ) A = C(X) ) then if A is an SV-ring then it has finite rank. Showing that this latter holds makes use of the theory of finite-valued lattice-ordered (abelian) groups. These two kinds of rings are investigated with an emphasis on the uniformly complete case. Fairly powerful machinery seems to have to be used, and even then, we do not know if there is a compact space X of finite rank that fails to be an SV-space.

Rings with finitely many nilpotent elements

It is well-known that a ring with no nonzero nilpotent elements - a so-called reduced ring - is a subdirect product of domains. Moreover, as we have recently shown [2], a prime ring with only finitely many nilpotent elements is either a domain or is finite. In view of these results, it is natural to ask what can be said in general about rings with only finitefy many nilpotent elements. A crucial property of such rings is that they contain no infinite zero subrings, hence we are led to consider rings with this property also. Our principal result is that for any ring R with only finitely many nilpotent elements is a direct sum of a reduced ring and a finite ring, where p(R) denotes the prime radical of R. One consequence is a finiteness theorem for periodic rings; another is the rather surprising result that every ring with infinitely many nilpotent elements has an infinite zero subring.