Varieties Obeying Homotopy Laws

Taylor, Walter

doi:10.4153/cjm-1977-054-9

Cited by 79 publications

(63 citation statements)

References 17 publications

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“…Larose and Zádori [28] recognized that a longstanding theorem due to Taylor [47] provided a term condition which could be used to restate the above conjecture.…”

Section: The Algebraic Dichotomy Conjecturementioning

confidence: 99%

Commutative idempotent groupoids and the constraint satisfaction problem

Bergman

Failing

2015

Algebra Univers.

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A restatement of the Algebraic Dichotomy Conjecture, due to Maroti and McKenzie, postulates that if a finite algebra A possesses a weak near-unanimity term, then the corresponding constraint satisfaction problem is tractable. A binary operation is weak near-unanimity if and only if it is both commutative and idempotent. Thus if the dichotomy conjecture is true, any finite commutative, idempotent groupoid (CI groupoid) will be tractable. It is known that every semilattice (i.e., an associative CI groupoid) is tractable. A groupoid identity is of Bol-Moufang type if the same three variables appear on either side, one of the variables is repeated, the remaining two variables appear once, and the variables appear in the same order on either side (for example, x(x(yz)) ≈ (x(xy))z). These identities can be thought of as generalizations of associativity. We show that there are exactly 8 varieties of CI groupoids defined by a single additional identity of Bol-Moufang type, derive some of their important structural properties, and use that structure theory to show that 7 of the varieties are tractable. We also characterize the finite members of the variety of CI groupoids satisfying the self-distributive law x(yz) ≈ (xy)(xz), and show that they are tractable.

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“…Larose and Zádori [28] recognized that a longstanding theorem due to Taylor [47] provided a term condition which could be used to restate the above conjecture.…”

Section: The Algebraic Dichotomy Conjecturementioning

confidence: 99%

Commutative idempotent groupoids and the constraint satisfaction problem

Bergman

Failing

2015

Algebra Univers.

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“…, y n ∈ {x, y} and x i = x, y i = y. The existence of a Taylor term is the weakest non-trivial idempotent Mal'tsev condition [20]. Taylor terms recently receive a great deal of attention because of the fact that they are conjectured to distinguish between tractable and NP-complete CSPs [6].…”

Section: Taylor Termmentioning

confidence: 99%

Mal’tsev conditions, lack of absorption, and solvability

2015

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Abstract. We provide a new characterization of several Mal'tsev conditions for locally finite varieties using hereditary term properties. We show a particular example how lack of absorption causes collapse in the Mal'tsev hierarchy, and point out a connection between solvability and lack of absorption. As a consequence, we provide a new and conceptually simple proof of a result of Hobby and McKenzie, saying that locally finite varieties with a Taylor term possess a term which is Mal'tsev on blocks of every solvable congruence in every finite algebra in the variety.

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“…Taylor proves in [7] that T\v E %v and the group laws obeyed by all groups in Tlv coincide with those laws obeyed by all groups in %v. Whether or not Tlv = §v remains an open question.…”

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confidence: 99%

“…Applications to Universal Algebra. Theorem 3.1 has apphcations to algebraic structures other than A-modules and, in fact, was inspired by Walter Taylor's paper Varieties obeying homotopy laws [7]. In this section we define some terms from Universal Algebra which we will use to describe these applications.…”

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confidence: 99%

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Fundamental groups of topological 𝑅-modules

Bateson¹

1982

Trans. Amer. Math. Soc.

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Abstract. The main result of this paper is that if R is a countable, Noetherian ring, then the underlying abelian group of every Ä-module is isomorphic to the fundamental group of some topological K-module. As a corollary, it is shown that for certain varieties V (e.g., varieties of finite type) every abelian group in V is isomorphic to the fundamental group of some arcwise connected topological algebra in V.

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Varieties Obeying Homotopy Laws

Cited by 79 publications

References 17 publications

Commutative idempotent groupoids and the constraint satisfaction problem

Commutative idempotent groupoids and the constraint satisfaction problem

Mal’tsev conditions, lack of absorption, and solvability

Fundamental groups of topological 𝑅-modules

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