Optimal strong Mal’cev conditions for omitting type 1 in locally finite varieties

Kearnes, Keith A.; Marković, Petar; McKenzie, Ralph

doi:10.1007/s00012-014-0289-9

Cited by 34 publications

(56 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following is an example of a locally finite variety V that omits the unary type but has no Taylor term of arity less than 4, proving that the arity in Corollary 2.2 is optimal. This is also observed in [18]. …”

Section: Strong Maltsev Conditionssupporting

confidence: 69%

See 1 more Smart Citation

Characterizations of several Maltsev conditions

et al. 2015

View full text Add to dashboard Cite

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, 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.

show abstract

Section: Strong Maltsev Conditionssupporting

confidence: 69%

“…Shortly after the announcement of Siggers's result, it was noted by Kearnes, Marković, and McKenzie [18] that one could replace the 6-ary term of Siggers by one of several types of 4-ary terms. Their proof employs a deep result of Barto, Niven, and the first author [5] on the complexity of the graph homomorphism problem.…”

Section: Strong Maltsev Conditionsmentioning

confidence: 99%

Characterizations of several Maltsev conditions

et al. 2015

View full text Add to dashboard Cite

show abstract

“…Theorem The following are equivalent for each finite idempotent algebra

boldA

boldA

is a Taylor algebra. For some

n ⩾ 2

boldA

has a term

t

of arity

n

that satisfies

t (x, x, \dots, x, y) \approx t (x, \dots, x, y, x) \approx \dots \approx t (x, y, x \dots, x) \approx t (y, x, \dots, x, x)

( weak NU term of arity

n

, or

n

–WNU for short). For each prime

n > | A |

boldA

has a term

t

of arity

n

that satisfies

t false(x_{1}, x_{2}, \dots, x_{n} false) \approx t false(x_{2}, \dots, x_{n}, x_{1} false)

( cyclic term ).

boldA

has a 6‐ary term

t

that satisfies

s (x, y, x, z, y, z) \approx s (y, x, z, x, z, y)

; ( 6‐ary Siggers term

s

boldA

has a 4‐ary term

…”

Section: Taylor Algebrasmentioning

confidence: 99%

“…Is there a common generalization? A particular interesting question is whether it is possible to further improve the weak 3‐cube term to the so‐called weak 3‐edge term . Open problem Does every idempotent Taylor algebra have 4‐ary term

e

satisfying the equations

e (y, y, x, x) \approx e (y, x, y, x) \approx e (x, x, x, y) ?

…”

Section: Open Problemsmentioning

confidence: 99%

“…Under this assumption, it is known that the CSP is NP‐complete whenever

boldA

is not Taylor and the algebraic dichotomy conjecture , confirmed in many special cases, states that the CSP is solvable in polynomial time otherwise. An intensive research motivated by this conjecture has brought a number of strong characterizations of finite Taylor algebras, including the equational condition given by Siggers refined by Kearnes, Marković and McKenzie : A finite algebra

boldA

is Taylor if and only if

boldA

has a 4‐ary idempotent term

s

satisfying the equation

s (r, a, r, e) \approx s (a, r, e, a) .

The fact that there is a weakest idempotent equational condition for finite algebras was unexpected and possible extension to at least some classes of infinite algebras was not considered until much later, in the context of infinite domain CSPs.…”