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Orderincompleteness and finite lambda reduction models
 Theoretical Computer Science
, 2003
"... Abstract Many familiar models of the untyped lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of fi or fijconversion, the pa ..."
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Abstract Many familiar models of the untyped lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of fi or fijconversion, the partial order is trivial on term denotations. Equivalently, theopen and closed term algebras of the untyped lambda calculus cannot be nontrivially partially ordered. Our second result is a syntactical characterization, in terms of socalled generalized Mal'cev operators, of those lambda theorieswhich cannot be induced by any nontrivially partially ordered model. We also consider a notion of finite models for the untyped lambda calculus, or more precisely, finite models of reduction. We demonstrate how such models can beused as practical tools for giving finitary proofs of term inequalities. 1 Introduction Perhaps the most important contribution in the area of mathematical programming semantics was the discovery, byD. Scott in the late 1960's, that models for the untyped lambda calculus could be obtained by a combination of ordertheoretic and topological methods. A long tradition of research in domain theory ensued, and Scott's methods havebeen successfully applied to many aspects of programming semantics.
On the complexity of some Maltsev conditions
, 2006
"... Abstract. This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but th ..."
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Abstract. This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but that in general these problems are EXPTIME complete. In addition, we provide sharp bounds in terms of the size of twogenerated free algebras on the number of terms needed to witness various Maltsev conditions, such as congruence distributivity. 1.
Axiomatizable and Nonaxiomatizable Congruence Prevarieties, Algebra Universalis
"... Abstract. If V is a variety of algebras, let L(V) denote the prevariety of all lattices embeddable in congruence lattices of algebras in V. We give some criteria for the firstorder axiomatizability or nonaxiomatizability of L(V). One corollary to our results is a nonconstructive proof that every co ..."
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Cited by 3 (2 self)
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Abstract. If V is a variety of algebras, let L(V) denote the prevariety of all lattices embeddable in congruence lattices of algebras in V. We give some criteria for the firstorder axiomatizability or nonaxiomatizability of L(V). One corollary to our results is a nonconstructive proof that every congruence npermutable variety satisfies a nontrivial congruence identity. 1.
CHARACTERIZATIONS OF SEVERAL MALTSEV CONDITIONS
"... Abstract. Tame congruence theory identifies six Maltsev conditions associated with locally finite varieties omitting certain types of local behaviour. Extending a result of Siggers, we show that of these six Maltsev conditions only two of them are equivalent to strong Maltsev conditions for locally ..."
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Abstract. Tame congruence theory identifies six Maltsev conditions associated with locally finite varieties omitting certain types of local behaviour. Extending a result of Siggers, we show that of these six Maltsev conditions only two of them are equivalent to strong Maltsev conditions for locally finite varieties. Besides omitting the unary type [24], the only other of these conditions that is strong is that of omitting the unary and affine types. We also provide novel presentations of some of the above Maltsev conditions. 1.
IDEMPOTENT nPERMUTABLE VARIETIES
"... Abstract. One of the important classes of varieties identified in tame congruence theory is the class of varieties which are npermutable for some n. In this paper we prove two results: (1) For every n> 1 there is a polynomialtime algorithm which, given a finite idempotent algebra A in a finite lan ..."
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Abstract. One of the important classes of varieties identified in tame congruence theory is the class of varieties which are npermutable for some n. In this paper we prove two results: (1) For every n> 1 there is a polynomialtime algorithm which, given a finite idempotent algebra A in a finite language, determines whether the variety generated by A is npermutable; (2) A variety is npermutable for some n if and only if it interprets an idempotent variety which is not interpretable in the variety of distributive lattices. 1.
Revised Manuscript 1 2 3 4 5 6 7 8
"... Abstract. We describe an easy way to determine whether the realization of a set of idempotent identities guarantees congruence modularity or the satisfaction of a nontrivial congruence identity. Our results yield slight strengthenings of Day’s Theorem and Gumm’s Theorem, which each characterize cong ..."
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Abstract. We describe an easy way to determine whether the realization of a set of idempotent identities guarantees congruence modularity or the satisfaction of a nontrivial congruence identity. Our results yield slight strengthenings of Day’s Theorem and Gumm’s Theorem, which each characterize congruence modularity. 1.
NOTES ON CONGRUENCE nPERMUTABILITY AND SEMIDISTRIBUTIVITY
"... T. Dent, K. Kearnes and Á. Szendrei define the derivative, Σ ′ , of a set of equations Σ and show, for idempotent Σ, that Σ implies congruence modularity if Σ ′ is inconsistent (Σ ′  = x ≈ y). In this paper we investigate other types of derivatives that give similar results for congruence npermuta ..."
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T. Dent, K. Kearnes and Á. Szendrei define the derivative, Σ ′ , of a set of equations Σ and show, for idempotent Σ, that Σ implies congruence modularity if Σ ′ is inconsistent (Σ ′  = x ≈ y). In this paper we investigate other types of derivatives that give similar results for congruence npermutable for some n, and for congruence semidistributivity. In a recent paper [1] T. Dent, K. Kearnes and Á. Szendrei study Maltsev conditions which imply congruence modularity from the point of view of the equations. Given a set of equations Σ they define the derivative Σ ′ of Σ. (The definition is given below.) Σ ′ ⊇ Σ and if Σ is idempotent then, if Σ ′ is inconsistent (that is, Σ ′  = x ≈ y), then any variety V that realizes Σ (each function symbol in Σ can be interpreted as a term of V such that the equations of Σ are satisfied) is congruence modular. While the converse is not true in general, they show that it is true if Σ is a set of linear, idempotent equations. So in particular if Σ consists of Day’s equations (for a fixed n) or Gumm’s equations, then Σ ′ is inconsistent. They use these results to prove interesting new results and give easy proofs of several existing theorems. One nice example: one of the equations in Day’s characterization of congruence modularity involves three variables: mi(x, u, u, y) ≈ mi+1(x, u, u, y) for i odd J. B. Nation wondered if there is a twovariable condition implying congruece modularity. In [9] he showed that this is the case. Using their results, the authors of [1] show that the above equation can be replaced by either or mi(x, x, x, y) ≈ mi+1(x, x, x, y)
SEMILATTICES WITH SECTIONALLY ANTITONE BIJECTIONS
"... Abstract. We study ∨semilattices with the greatest element 1 where on each interval [a,1] an antitone bijection is defined. We characterize these semilattices by means of two induced binary operations proving that the resulting algebras form a variety. The congruence properties of this variety and ..."
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Abstract. We study ∨semilattices with the greatest element 1 where on each interval [a,1] an antitone bijection is defined. We characterize these semilattices by means of two induced binary operations proving that the resulting algebras form a variety. The congruence properties of this variety and the properties of the underlying semilattices are investigated. We show that this variety contains a single minimal subquasivariety.