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We generalise in three different directions two well-known results in universal algebra. Gratzer, Lakser and Plonka proved that independent subvarieties $${\mathcal{V}_{1}, \mathcal{V}_{2}}$$ of a variety $${\mathcal{V}}$$ are disjoint and such that their join $${\mathcal{V}_{1} \vee \mathcal{V}_{2}}$$ (in the lattice of subvarieties of $${\mathcal{V}}$$ ) is their direct product $${\mathcal{V}_{1} \times \mathcal{V}_{2}}$$ . Jonsson and Tsinakis provided a partial converse to this result: if $${\mathcal{V}}$$ is congruence permutable and $${\mathcal{V}_{1}, \mathcal{V}_{2}}$$ are disjoint, then they are independent (and so $${\mathcal{V}_{1} \vee \mathcal{V}_{2} = \mathcal{V}_{1} \times \mathcal{V}_{2}}$$ ). We show that (i) if $${\mathcal{V}}$$ is subtractive, then Jonsson’s and Tsinakis’ result holds under some minimal assumptions; (ii) if $${\mathcal{V}}$$ satisfies some weakened permutability conditions, then disjointness implies a generalised notion of independence and $${\mathcal{V}_{1} \vee \mathcal{V}_{2}}$$ is the subdirect product of $${\mathcal{V}_{1}}$$ and $${\mathcal{V}_2}$$ ; (iii) the same holds if $${\mathcal{V}}$$ is congruence 3-permutable.

The authors’ description of the inverse semigroups S for which the lattice ℒℱ(S) of full inverse subsemigroups is lower semimodular is used to describe those for which (a) the lattice ℒ(S) of all inverse subsemigroups or (b) the lattice 𝒞o(S) of convex inverse subsemigroups has that property. In each case, we show that this occurs if and only if the entire lattice is a subdirect product of ℒℱ(S) with ℒ(E S ), or 𝒞o(E S ), respectively, where E S is the semilattice of idempotents of S; a simple necessary and sufficient condition is found for each decomposition. For a semilattice E, ℒ(E) is in fact always lower semimodular, and 𝒞o(E) is lower semimodular if and only if E is a tree. The conjunction of these results leads to quite a divergence between the ultimate descriptions in the two cases, ℒ(S) and 𝒞o(S), with the latter being substantially richer.

We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of MBA's as powerset algebras of certain directed graphs with a set of "marked" points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly $${{2^{{{3.2}^r}.2^{2^r - 1}}}}$$ elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r.

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated ``subdirect product'' as introduced by McMullen (1976); it is dual to the ``wreath product'' construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces ``of unusually large genus'' that first appeared in works by Coxeter (1937), Ringel (1956), and McMullen, Schulz, and Wills (1983). Via ``projections of deformed wedge products'' we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in R^3. As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations (``moduli'') for the surfaces in R^3. In order to prove that there are many moduli, we introduce the concept of ``affine support sets'' in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in R^3 via dual 4-polytopes., Comment: 17 pages, 5 figures

A nearlattice A is a meet semilattice in which every initial segment is a lattice; A is said to be Boolean if all of these lattices are Boolean. Overriding on A is a binary operation \(\triangleleft\) defined by \( a \triangleleft {b} := \sup\{x\colon (x \le a \mbox{ and } x {\mathrel{\mbox{\,\raisebox{.7ex}{$\scriptstyle|$}\hspace{-.75ex}\raisebox{-.35ex}{$\circ$}}}}b) \mbox{ or } x \le b\} \), where \(x{\mathrel{\mbox{\,\raisebox{.7ex}{$\scriptstyle|$}\hspace{-.75ex}\raisebox{-.35ex}{$\circ$}}}}b\) means that the join of x and b exists. In particular, \({\triangleleft}\) extends the partial join operation. Natural examples of overriding nearlattices are provided by trees and by appropriate algebras of partial functions. In the paper, elementary properties of overriding are studied, an axiomatic description of it is obtained, and several results concerning associativity of the operation and the structure of function nearlattices with overriding are presented. The main result is a representation theorem (in terms of algebras of partial functions) for Boolean nearlattices with associative overriding.

An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form avariety SCIL, which is not locally finite, but it is proved that SCIL has the finite embeddability property (FEP). More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains.

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In a previous paper, we give a construction G(M) for a graph with prescribed endomorphism monoid M known as a [Formula: see text]-graph. Using this construction, we derived bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. In this paper, we generalize the [Formula: see text]-graph construction and derive several new bounds for monoid classes not handled by our first paper. Among these are the so called "strong semilattices of C-semigroups" where C is one of the following: Groups, Abelian Groups, Rectangular Groups, and completely simple semigroups.

In 1987, Romanowska and Roszkowska found an explicit description of the lattice of varieties of differential groupoids. In 2008, the author proved that the lattice of quasivarieties of differential groupoids is \( \mathcal{Q} \)-universal. In the present article, we find an example of a minimal \( \mathcal{Q} \)-universal variety of differential groupoids.

A congruence preserving function on a subdirect product of two finite Mal’cev algebras is polynomial if it induces polynomial functions on the subdirect factors and there are no skew congruences between the projection kernels. As a special case, if the direct product A × B of finite algebras A and B in a congruence permutable variety has no skew congruences, then the polynomial functions on A × B are exactly direct products of polynomials on A and on B. These descriptions apply in particular to classical polynomial functions on nonassociative rings. Also, for finite algebras A, B in a variety with majority term, the polynomial functions on A × B are exactly the direct products of polynomials on A and on B. However in arbitrary congruence distributive varieties the corresponding result is not true.

We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup., Comment: 29 pages, no figures

A generalized MV-algebra A is called representable if it is a subdirect product of linearly ordered generalized MV-algebras. Let S be the system of all congruence relations ϱ on A such that the quotient algebra A/ϱ is representable. In the present paper we prove that the system S has a least element.

This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices ${\mathbf L}$ easily reduces to the case when ${\mathbf L}$ is a subdirect product of two simple lattices ${\mathbf L}_1$ and ${\mathbf L}_2$. Our main result claims that such a lattice is locally order affine complete iff ${\mathbf L}_1$ and ${\mathbf L}_2$ are tolerance trivial and one of the following three cases occurs: 1) ${\mathbf L}={\mathbf L}_1\times {\mathbf L}_2$, 2) ${\mathbf L}$ is a maximal sublattice of the direct product, 3) ${\mathbf L}$ is the intersection of two maximal sublattices, one containing $\langle 0,1\rangle $ and the other $\langle 1,0\rangle $.

The notion of internal subdirect decomposition can be defined in each variety of algebras. In the present note we prove the validity of a cancellation rule concerning such decompositions for lattice ordered groups and for GMV-algebras. For the case of groups, this cancellation rule fails to be valid.

J. Plonka in [12] noted that one could expect that the regularization ℛ(K) of a variety K of unary algebras is a subdirect product of K and the variety D of all discrete algebras (unary semilattices), but is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties K which are contained in the generalized variety TDir of the so-called trap-directable algebras.

We introduce perfect effect algebras and we show that every perfect algebra is an interval in the lexicographical product of the group of all integers with an Abelian directed interpolation po-group. To show this we introduce prime ideals of effect algebras with the Riesz decomposition property (RDP). We show that the category of perfect effect algebras is categorically equivalent to the category of Abelian directed interpolation po-groups. Moreover, we prove that any perfect effect algebra is a subdirect product of antilattice effect algebras with the RDP.

In this paper we describe the covers of a left ample semigroup that arise from strict (2,1)-embeddings in left factorizable left ample monoids.

Subdirectly irreducible acts and finitely subdirectly irreducible acts are characterized, for acts over a commutative semigroup that satisfy finiteness conditions.

In this paper, we consider the lattice Subf S of full subsemigroups of an inverse semigroup S. Our first main theorem states that for any inverse semigroup S, Subf S is a subdirect product of the lattices of full subsemigroups of its principal factors, so that Subf S is distributive [meet semidistributive, join semidistributive, modular, semimodular] if and only if the lattice of full subsemigroups of each principal factor is. To examine such inverse semigroups, therefore, we need essentially only consider those which are 0-simple. For a 0-simple inverse semigroup S (not a group with zero), we show that in fact each of modularity, meet semidistributivity and join semidistributivity of Subf S is equivalent to distributivity of S, that is, S is the combinatorial Brandt semigroup with exactly two nonzero idempotents and two nonidempotents. About semimodularity, however, we concentrate only on the completely 0-simple case, that is, Brandt semigroups. For a Brandt semigroup S (not a group with zero), semimodularity of Subf S is equivalent to distributivity of Subf S. Finally, we characterize an inverse semigroup S for which Subf S is a chain.

Let U be the variety of idempotent semirings satisfying xy + yx = yx + xy. We give a description of the lattice L(U) of subvarieties of U: L(U) happens to be a 662-element distributive lattice which is isomorphic to a subdirect product of the lattices L(S+ l) and L(S. l), where L(S+ l) [L(S. l)] denotes the variety of all idempotent semirings whose additive [multiplicative] reduct is a semilattice. In particular, U = L(S+ l) ⋁ L(S. l). Every subvariety of U is finitely generated and finitely based. If S ∈ U, then both the additive reduct and the multiplicative reduct of S are regular bands. The structural relevance of the least U-congruence is investigated.

The additive reduct of a band semiring is a regular band. We refer to [1] and [2] for a general background on semigroups, and on bands in particular. A semiring (S,+, ·) is said to be an idempotent semiring if both the reducts (S,+) and (S, ·) are bands. Green’s D [L-, R-] relation on the additive reduct (S,+) will be denoted by + D [ + L , + R ]. It is easy to see that if (S,+, ·) is an idempotent semiring, then + D is a semiring congruence, whereas + L induces a congruence on the multiplicative reduct (S, ·) (see also [3]). An idempotent semiring (S,+, ·) is said to be a band semiring if + D is the least (distributive) lattice congruence on S , or equivalently, if S satisfies the identities x+ xy + x ≈ x ≈ x+ yx+ x (see Theorem 2.2 of [3]). We shall show that the additive reduct of a band semiring is a regular band. We shall use the fact that a band is regular if and only if the Green’s relations L and R are congruence relations (see Section II.3 of [2]). Lemma. Let (S,+) be a band. The Green’s relation L is not a congrence relation on S if and only if S contains either a subband isomorphic to E1 or ∗The research is supported by an NSF(China) grant #10471112. The research of the first author is supported by a grant of the Youth Scintific Research Foundation of Southwest Normal University (#SWNUQ2004003). 440 Wang, Zhou, and Guo a subband isomorphic to E2 , where E1 and E2 are respectively given by + a b c d e a a a c d e b b b c d e c d e c d e d d d c d e e e e c d e and + a b c d e f g h a a a c d e c d e b b b f g h f g h c d e c d e c d e d d d c d e c d e e e e c d e c d e f g h f g h f g h g g g f g h f g h h h h f g h f g h . Proof. The Green’s relation L on the band S is not a congruence if and only if there exist a, b, k ∈ S such that aLb but k+a is not L-related to k+ b . If this is the case, then with c = a + k , the subband of S generated by a, b and c is isomorphic to E1 if c = b + c , and is isomorphic to E2 if c = b + c . Conversely, if S contains a subband isomorphic to E1 or E2 , then obviously the L-relation is not a congruence on S . Theorem. The additive reduct of a band semiring is a regular band. Proof. Let (S,+, ·) be a band semiring. Using the Lemma and duality, it suffices to show that the band (S,+) cannot contain a subband isomorphic to E1 or E2 . (1) Suppose that (S,+) has a subband T isomorphic to E1 . We identify T with E1 . First we have a = a+ ac+ a = a(a+ c+ a) = ad. Since a + Lb and + L is a congruence on (S, ·), we have a + Lab . Then a = ad = a(b+ d) = ab+ ad = ab+ a = ab. Since + D is a lattice congruence we have that ab + Dcb in S . So we have b = b+ db+ b = (b+ d+ b)b = db = (a+ c+ a)b = ab+ cb+ ab = ab = a, a contradiction. Therefore (S,+) has no subband isomorphic to E1 . (2) Suppose S has a subband T isomorphic to E2 and we identify T with E2 . First we have a = a+ ae+ a = a(a+ e+ a) = ae and b = b+ gb+ b = (b+ g + b)b = gb. Wang, Zhou, and Guo 441 Since + D is a lattice congruence, we obtain that ab + Ddb + Dah . So we have ab = ab+ db+ ab = (a+ d+ a)b = db and ab = ab+ ah+ ab = a(b+ h+ b) = ah. Thus, noticing that ac + Dcb + Dab , we have (c+ a)(c+ b) = (c+ a)c+ (c+ a)b = c+ ac+ cb+ ab = c+ (ac+ ab) = c+ a(c+ b) = c+ ae = c+ a = d and (c+ a)(c+ b) = c(c+ b) + a(c+ b) = c+ cb+ ac+ ab = c+ (cb+ ab) = c+ (c+ a)b = c+ db = c+ ab. Therefore we have c + ab = d . Similarly, applying the distributive law to (f+a)(f+b), one can get f+ab = h in combination with b = gb and ab = ah . Thus we obtain that d = c+ ab = (a+ f) + ab = a+ (f + ab) = a+ h = e, a contradiction. So S has no subband isomorphic to E2 either. Thus the proof is complete. Note 1. Let (S, ·) be a band and define an addition by: for a, b ∈ S, a = a + b . Then (S,+, ·) is a band semiring. Thus, any band is isomorphic to the multiplicative reduct of some band semiring. Note 2. The additive reduct of a band semiring need not be a normal band: see the example following Construction 3.5 of [3]. Note 3. For a band semiring S , the relation + L is a semiring congruence. From this and its dual it follows that S is a subdirect product of the semirings S/ + L and S/ + R . 442 Wang, Zhou, and Guo Acknowledgment The authors would like to thank Professor Francis Pastijn for his valuable suggestions and help. References [1] Howie, J. M., “Fundamentals of Semigroup Theory”, Clarendon Press, Ox- ford, 1995. [2] Petrich, M., “Lectures in Semigroups”, Akad. Berlag, Berlin, 1977. [3] Sen, M. K., Guo, Y. Q. and Shum, K. P., A class of idempotent semirings, Semigroup Forum 60 (2000), 351–367. Department of Mathematics Harbin Institute of Technology Harbin (150001), China Department of Mathematics Southwest China University Chongqing (400715), China zpwang@swu.edu.cn Department of Mathematics Jiangxi Normal University Nanchang (330027), China ylzhou185@163.com Department of Mathematics Southwest China University Chongqing (400715) China yqguo259@swu.edu.cn Received November 7, 2004 and in final form August 19, 2005 Online publication January 23, 2006

The aim of this paper is to introduce the notions of fuzzy filters, fuzzy prime filters and the cosets of a fuzzy filter in BL-algebras and investigate some of their properties. The fuzzy filter generated by a fuzzy set is discussed. Some characterizations of fuzzy filters and fuzzy prime filters are derived. The extension theorem of fuzzy prime filters and the fuzzy prime filters theorem are established. Finally, we prove that the algebra L/f which is the set of all cosets of f is a BL-algebra, and is isomorphic to the BL-algebra L/ff(1), where ff(1) = {x ∈ L|f(x) =f(1)}. Moreover, each BL-algebra L is a subdirect product of linearly ordered BL-algebras L/fi (i ∈ Γ). where fi (i ∈ Γ) are fuzzy prime filters of L.

Pseudo-effect algebras are partial algebras (E;+,0,1) with a partially defined addition + which is not necessarily commutative and therefore with two complements, left and right. If they satisfy a special kind of the Riesz decomposition property, they are intervals in unital po-groups. The general comparability property in unital po-groups with strong unit (G,u), allows to compare elements of G in some intervals with Boolean ends. Such a po-group is always an l-group admitting a state. We prove that every such (G,u) is a subdirect product of linearly ordered unital po-groups.

In this paper, we introduce p-ideals in semirings. A new form of regularity, which is compatible with p-ideals, is defined. Our aim is to explore the possibilities of establishing an ideal theory in semirings, going alongside the existing literature of semiring theory.

Let [Formula: see text] be the prime graph associated with a finite group [Formula: see text] and [Formula: see text] be the degree pattern of [Formula: see text]. A finite group [Formula: see text] is said to be [Formula: see text]-fold [Formula: see text]-characterizable if there exist exactly [Formula: see text] nonisomorphic groups [Formula: see text] such that [Formula: see text] and [Formula: see text]. The purpose of this paper is two-fold. First, it shows that the symmetric group [Formula: see text] is [Formula: see text]-fold [Formula: see text]-charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, [Formula: see text] and [Formula: see text], which are [Formula: see text]-fold [Formula: see text]-characterizable with [Formula: see text].

We introduce a product on an effect algebra. We prove that every product effect algebra with the Riesz decomposition property (RDP), is an interval in an Abelian unital interpolation po-ring, and we show that the category of product effect algebras with the RDP is categorically equivalent with the category of unital Abelian interpolation po-rings. In addition, we show that every product effect algebra with the RDP and with 1 as a product unity is a subdirect product of antilattice product effect algebras with the RDP.

i>BL-algebras are the Lindenbaum algebras of the propositional calculus coming from the continuous triangular norms and their residua in the real unit interval. Any i>BL-algebra is a subdirect product of local (linear) i>BL-algebras. A local i>BL-algebra is either locally finite (and hence an i>MV-algebra) or perfect or peculiar. Here we study extensively perfect i>BL-algebras characterizing, with a finite scheme of equations, the generated variety. We first establish some results for general i>BL-algebras, afterwards the variety is studied in detail. All the results are parallel to those ones already existing in the theory of perfect i>MV-algebras, but these results must be reformulated and reproved in a different way, because the axioms of i>BL-algebras are obviously weaker than those for i>MV-algebras.

An algebra A is said to be a congruence-preserving extension of a subalgebra B if the mapping from the congruence lattice of B to that of A, assigning to each congruence relation β on B the minimal congruence relation on A containing β, is an isomorphism. We give a necessary and sufficient condition on the congruence lattice of a subdirect product B of finitely many algebras in a congruence-distributive variety that the full direct product be a congruence-preserving extension of B. We give several applications to congruence lattices of lattices.

In this paper, we investigate the comparability structure over exchange rings. It is shown that the subdirect product of an exchange ring with stable range one and an exchange ring satisfying the comparability is also an exchange ring satisfying the comparability. This provides a new class of exchange rings satisfying the comparability. Furthermore, we investigate the s-comparability over exchange rings. This generalizes the corresponding results of Goodearl and Chen.

A subsemigroup S of a semigroup Q is an order in Q if for every \(\) there exist \(\) such that \(\), where a and d are contained in (maximal) subgroups of Q, and \(\) and \(\) are their inverses in these subgroups. A regular semigroup S is strict if it is a subdirect product of completely (0-)simple semigroups.

We propound an approach through which techniques of the theory of quasivarieties of predicate systems are brought to bear on partial algebras. For every partial algebra A, two predicate representations are treated. The first is the graph of A whose basic operations are graphs of the basic operations on A. The second representation results from the graph of A by adding domains of the operations on A to its basic relations. Studying partial algebras from various perspectives makes it necessary to deal with different equality semantics. Here we present a general definition of semantics that stretches over such instances as weak semantics, Evans’ semantics. Kleene semantics, and strong semantics. On a set of all semantics, the preorder is induced in increasing “force,” and it is proved that certain of the properties of varieties of partial algebras in a given semantics are individuated by the position it takes in that set. We argue that every variety of partial algebras, in any semantics, is in correspondence with a Horn class of predicate systems which admits a generation operator and is closed under direct limits and retracts. For such classes we prove analogs of the Birkhoff theorem on subdirect decompositions and of the Taylor theorem on residual smallness. Therefore, these are also applicable to varieties of partial algebras in arbitrary semantics.

In the paper we obtain a new characterization of the BCK-algebras which are subdirect product of BCK-chains. We give an axiomatic algebraizable extension of the BCK-calculus, by means of a recursively enumerable set of axioms, such that its equivalent algebraic semantics is definitionally equivalent to the quasivariety of BCK-algebras generated by the BCK-chains. We propose the concept of "linearization of a system" and we give some examples.

We study Dedekind complete commutative BCK-algebras with the relative cancellation property and their connection with corresponding universal groups. We shall characterize Dedekind orthogonally complete atomic and Archimedean BCK-algebras, generalizing results of Jakubik known for MV-algebras. Finally, we characterize those Dedekind complete and atomic commutative BCK-algebras that are isomorphic to direct products of basic BCK-chains, generalizing a result of Cignoli for MV-algebras.

A solution is given for the word problem for free idempotent distributive semirings. Using this solution the latticeL (ID) of subvarieties of the variety ID of idempotent distributive semirings is determined. It turns out thatL (ID) is isomorphic to the direct product of a four-element lattice and a lattice which is itself a subdirect product of four copies of the latticeL(B) of all band varieties. ThereforeL(ID) is countably infinite and distributive. Every subvariety of ID is finitely based.

inversive semiring and a Clifford semiring and show that a semiring S is a subdirect product of a distributive lattice and a ring if and only if S is an E-inversive strong distributive lattice of halfrings. Further a Clifford semiring which is, in fact, an inversive subdirect product of a distributive lattice and a ring, is characterized as a strong distributive lattice of rings. Finally, as a consequence of these results we extend a result of Galbiati and Veronesi [2] in the case of Boolean semirings.

It is shown that any n-chromatic graph is a full subdirect product of copies of the complete graphs K_n and K_{n+1} , except for some easily described graphs which are full subdirect products of copies of K_{n+1} − l°–°r and K_{n+2} − l°–°r; full means here that the projections of the decomposition are epimorphic in edges. This improves some results of Sabidussi. Subdirect powers of K_n or K_{n+1} − l°–°r are also characterized, and the subdirectly irreducibles of the quasivariety ofn -colorable graphs with respect to full and ordinary decompositions are determined.

For a given transitive binary relation e on a set E, the transitive closures of open (i.e., co-transitive in e) sets, called the regular closed subsets, form an ortholattice Reg(e), the extended permutohedron on e. This construction, which contains the poset Clop(e) of all clopen sets, is a common generalization of known notions such as the generalized permutohedron on a partially ordered set on the one hand, and the bipartition lattice on a set on the other hand. We obtain a precise description of the completely join-irreducible (resp., completely meet-irreducible) elements of Reg(e) and the arrow relations between them. In particular, we prove that (1) Reg(e) is the Dedekind-MacNeille completion of the poset Clop(e); (2) Every open subset of e is a set-theoretic union of completely join-irreducible clopen subsets of e; (3) Clop(e) is a lattice iiff every regular closed subset of e is clopen, iff e contains no "square" configuration, iff Reg(e)=Clop(e); (4) If e is finite, then Reg(e) is pseudocomplemented iff it is semidistributive, iff it is a bounded homomorphic image of a free lattice, iff e is a disjoint sum of antisymmetric transitive relations and two-element full relations. We illustrate the strength of our results by proving that, for n greater than or equal to 3, the congruence lattice of the lattice Bip(n) of all bipartitions of an n-element set is obtained by adding a new top element to a Boolean lattice with n2^{n-1} atoms. We also determine the factors of the minimal subdirect decomposition of Bip(n)., 25 pages

For each natural number $d$ we construct a $3$-generated group $H_d$, which is a subdirect product of free groups, such that the cohomological dimension of $H_d$ is $d$. Given a group $F$ and a normal subgroup $N \lhd F$ we prove that any right angled Artin group containing the special HNN-extension of $F$ with respect to $N$ must also contain $F/N$. We apply this to construct, for every $d \in \mathbb{N}$, a $4$-generated group $G_d$, embeddable into a right angled Artin group, such that the cohomological dimension of $G_d$ is $2$ but the cohomological dimension of any right angled Artin group, containing $G_d$, is at least $d$. These examples are used to show the non-existence of certain "universal" right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular we show that for every $n\in \mathbb{N}$ there exists $\delta(n) \in \mathbb{N}$ such that any $n$-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the $\delta(n)$-th direct power of the free group of rank $2$. As another corollary we derive that any $n$-generated finitely presented residually free group embeds into the direct product of at most $\delta(n)$ limit groups., Comment: v4: accepted in this format for publication in Intern. J. Algebra and Comput.; 12 pages

Global subdirect products of mutually disjoint algebras are exactly the algebras of global sections of sheaves [17]. We say that an algebra A is globally indecomposable if for every global subdirect product A1¤ {Ai : i I} such that A1 is isomorphic to A and I is a compact space we have that there exists i I such that pi : A1“Ai is an isomorphism, where pi is the canonical projection. For an algebra A, Con(A) denotes the congruence lattice of A and for a class M of algebras define S(A, M)= {u Con(A): A/u$B for some B in M}. In this paper we search the varieties of

We construct a countably infinite descending chain of Q-universal quasivarieties of graphs, starting with D, and show that its intersection has a distributive countably infinite lattice of subquasivarieties. It is also proved that the quasivariety of symmetric endographs is Q-universal.

Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group $G$, a set $I$ and with two bijections $\lambda,\rho:I \to I.$ Using a clever construction on the ordinal sum of $(G^+)^I$ and $(G^-)^I,$ we can define a pseudo effect algebra which can be non-commutative even if $G$ is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop., Comment: arXiv admin note: text overlap with arXiv:1306.0304

We show that for any tolerance $R$ on $U$, the ordered sets of lower and upper rough approximations determined by $R$ form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if $R$ is induced by an irredundant covering of $U$, and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set $\mathit{RS}$ of rough sets determined by a tolerance $R$ on $U$ is a complete lattice if and only if it is a complete subdirect product of the complete lattices of lower and upper rough approximations. We show that $R$ is a tolerance induced by an irredundant covering of $U$ if and only if $\mathit{RS}$ is an algebraic completely distributive lattice, and in such a situation a quasi-Nelson algebra can be defined on $\mathit{RS}$. We present necessary and sufficient conditions which guarantee that for a tolerance $R$ on $U$, the ordered set $\mathit{RS}_X$ is a lattice for all $X \subseteq U$, where $R_X$ denotes the restriction of $R$ to the set $X$ and $\mathit{RS}_X$ is the corresponding set of rough sets. We introduce the disjoint representation and the formal concept representation of rough sets, and show that they are Dedekind--MacNeille completions of $\mathit{RS}$., Revised version (28 pages, 1 figure)

This paper applies techniques of algebraic approximation to provide effective algorithms to determine the validity of universally quantified implications over lattice structures. We generalize the known result which states that any semilattice is approximated in the two element lattice. We show that the validity of a universally quantified implication ψ over a possibly infinite domain can be determined by examining its validity over a simpler domain the size of which is related to the number of constants in ψ. Both the known as well as the new results have high potential in providing practical automated techniques in various areas of application in computer science.

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.