Independence of algebras with edge term

In [A. L. Foster, The identities of — and unique subdirect factorization within — classes of universal algebras, Math. Z. 62 (1955) 171–188], two varieties [Formula: see text] of the same type are defined to be independent if there is a binary term [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give necessary and sufficient conditions for two finite algebras with a Mal’cev term (or, more generally, with an edge term) to generate independent varieties. In particular we show that the independence of finitely generated varieties with edge term can be decided by a polynomial time algorithm.

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Weight-Equitable Subdivision of Red and Blue Points in the Plane

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195918500024 ◽

2018 ◽

Vol 28 (01) ◽

pp. 39-56 ◽

Cited By ~ 2

Author(s):

Jude Buot ◽

Mikio Kano

Keyword(s):

General Position ◽

Sufficient Conditions ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Total Weight ◽

Blue Point ◽

Disjoint Sets ◽

Necessary And Sufficient ◽

Positive Integers ◽

Special Case

Let [Formula: see text] and [Formula: see text] be two disjoint sets of red points and blue points, respectively, in the plane in general position. Assign a weight [Formula: see text] to each red point and a weight [Formula: see text] to each blue point, where [Formula: see text] and [Formula: see text] are positive integers. Define the weight of a region in the plane as the sum of the weights of red and blue points in it. We give necessary and sufficient conditions for the existence of a line that bisects the weight of the plane whenever the total weight [Formula: see text] is [Formula: see text], for some integer [Formula: see text]. Moreover, we look closely into the special case where [Formula: see text] and [Formula: see text] since this case is important to generate a weight-equitable subdivision of the plane. Among other results, we show that for any configuration of [Formula: see text] with total weight [Formula: see text], for some integer [Formula: see text] and odd integer [Formula: see text], the plane can be subdivided into [Formula: see text] convex regions of weight [Formula: see text] if and only if [Formula: see text]. Using the proofs of the main result, we also give a polynomial time algorithm in finding a weight-equitable subdivision in the plane.

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BICYCLIC SUBSEMIGROUPS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s021819671000556x ◽

2010 ◽

Vol 20 (01) ◽

pp. 89-113 ◽

Cited By ~ 7

Author(s):

EMANUELE RODARO

Keyword(s):

Inverse Semigroup ◽

Free Product ◽

Sufficient Conditions ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Inverse Semigroups ◽

Amalgamated Free Product ◽

Free Product With Amalgamation ◽

Necessary And Sufficient ◽

Completely Semisimple

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max {|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the [Formula: see text]-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the [Formula: see text]-classes to be finite.

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FAST ISOMORPHISM TESTING IN ARITHMETICAL VARIETIES

International Journal of Algebra and Computation ◽

10.1142/s0218196703001572 ◽

2003 ◽

Vol 13 (04) ◽

pp. 499-506

Author(s):

TOMASZ A. GORAZD

Keyword(s):

Finite Fields ◽

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Boolean Algebras ◽

Heyting Algebras ◽

Subdirectly Irreducible ◽

Finite Algebras ◽

Element Chain ◽

Subdirectly Irreducible Algebras

Let [Formula: see text] be a finitely generated, arithmetical variety such that all subdirectly irreducible algebras from [Formula: see text] have linearly ordered congruences. We show that there is a polynomial time algorithm that tests the existing of an isomorphism between any two finite algebras from [Formula: see text]. This includes the following classical structures in algebra: • Boolean algebras. • Varieties of rings generated by finitely many finite fields. • Varieties of Heyting algebras generated by an n–element chain.

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Learning Importance of Preferences

10.29007/v68w ◽

2018 ◽

Author(s):

Ying Zhu ◽

Mirek Truszczynski

Keyword(s):

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Scoring Rules ◽

Np Hard ◽

Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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