POINTLIKE SETS, HYPERDECIDABILITY AND THE IDENTITY PROBLEM FOR FINITE SEMIGROUPS

In this paper, we give a relationship between the identity problem and the problem of deciding whether certain subsets of nilpotent semigroups are pointlike. We then use this to give an example of a pseudovariety which has a decidable membership problem, but for which one cannot decide pointlike sets. Then, by modifying the equations, we show that no graph is fundamentally hyperdecidable by constructing, for each graph, a labeling over a nilpotent semigroup for which we cannot decide inevitability with respect to the pseudovariety defined by these equations.

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Undecidability of the identity problem for finite semigroups

Journal of Symbolic Logic ◽

10.2307/2275184 ◽

1992 ◽

Vol 57 (1) ◽

pp. 179-192 ◽

Cited By ~ 33

Author(s):

Douglas Albert ◽

Robert Baldinger ◽

John Rhodes

Keyword(s):

Word Problem ◽

Finite Semigroup ◽

Identity Problem ◽

Finite Semigroups ◽

Or Groups ◽

Finite Set ◽

Variety Of Semigroups ◽

Finite Products

In 1947 E. Post [28] and A. A. Markov [18] independently proved the undecidability of the word problem (or the problem of deducibility of relations) for semigroups. In 1968 V. L. Murskil [23] proved the undecidability of the identity problem (or the problem of deducibility of identities) in semigroups.If we slightly generalize the statement of these results we can state many related results in the literature and state our new results proved here. Let V denote either a (Birkhoff) variety of semigroups or groups or a pseudovariety of finite semigroups. By a very well-known theorem a (Birkhoff) variety is defined by equations or equivalently closed under substructure, surmorphisms and all products; see [7]. It is also well known that V is a pseudovariety of finite semigroups iff V is closed under substructure, surmorphism and finite products, or, equivalently, determined eventually by equations w1 = w1′, w2 = w2′, w3 = w3′,… (where the finite semigroup S eventually satisfies these equations iff there exists an n, depending on S, such that S satisfies Wj = Wj′ for j ≥ n). See [8] and [29]. All semigroups form a variety while all finite semigroups form a pseudovariety.We now consider a table (see the next page). In it, for example, the box denoting the “word” (identity) problem for the psuedovariety V” means, given a finite set of relations (identities) E and a relation (identity) u = ν, the problem of whether it is decidable that E implies u = ν inside V.

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INTERPRETING GRAPH COLORABILITY IN FINITE SEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196706002846 ◽

2006 ◽

Vol 16 (01) ◽

pp. 119-140 ◽

Cited By ~ 20

Author(s):

MARCEL JACKSON ◽

RALPH McKENZIE

Keyword(s):

Membership Problem ◽

Finite Semigroups ◽

Variety Of Semigroups ◽

Membership Problems

We show that a number of natural membership problems for classes associated with finite semigroups are computationally difficult. In particular, we construct a 55-element semigroup S such that the finite membership problem for the variety of semigroups generated by S interprets the graph 3-colorability problem.

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The subpower membership problem for semigroups

International Journal of Algebra and Computation ◽

10.1142/s0218196716500612 ◽

2016 ◽

Vol 26 (07) ◽

pp. 1435-1451 ◽

Cited By ~ 4

Author(s):

Andrei Bulatov ◽

Marcin Kozik ◽

Peter Mayr ◽

Markus Steindl

Keyword(s):

Finite Semigroup ◽

Transformation Semigroup ◽

Membership Problem ◽

Clifford Semigroup ◽

Direct Power ◽

Nilpotent Semigroup ◽

Full Transformation ◽

Commutative Semigroups ◽

Np Complete ◽

A Finite Group

Fix a finite semigroup [Formula: see text] and let [Formula: see text] be tuples in a direct power [Formula: see text]. The subpower membership problem (SMP) asks whether [Formula: see text] can be generated by [Formula: see text]. If [Formula: see text] is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in [Formula: see text]. For semigroups this problem always lies in PSPACE. We show that the [Formula: see text] for a full transformation semigroup on [Formula: see text] or more letters is actually PSPACE-complete, while on [Formula: see text] letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup [Formula: see text] embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then [Formula: see text] is in P; otherwise it is NP-complete.

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THE SEMIDIRECTLY CLOSED PSEUDOVARIETY GENERATED BY APERIODIC BRANDT SEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s021819670100053x ◽

2001 ◽

Vol 11 (02) ◽

pp. 247-267 ◽

Cited By ~ 1

Author(s):

M. LURDES TEIXEIRA

Keyword(s):

Universal Algebra ◽

Brandt Semigroup ◽

Membership Problem ◽

Finite Semigroups ◽

Brandt Semigroups

This paper presents a study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B2, denoted V*(B2). We construct a basis of pseudoidentities for the semidirect powers of the pseudovariety generated by B2 which leads to the main result, which states that V*(B2) is decidable. Independently, using some suggestions given by J. Almeida in his book "Finite Semigroups and Universal Algebra", we constructed an algorithm to solve the membership problem in V* (B2).

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UNDECIDABILITY, AUTOMATA, AND PSEUDOVARITIES OF FINITE SEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196799000278 ◽

1999 ◽

Vol 09 (03n04) ◽

pp. 455-473 ◽

Cited By ~ 23

Author(s):

JOHN RHODES

Keyword(s):

Finite Number ◽

Finite Groups ◽

Membership Problem ◽

Joint Paper ◽

Finite Semigroups ◽

Membership Problems

The author proves for each of the operations # equalling *, °, **, □ or m, there exist pseudovarieties of finite semigroups [Formula: see text] and [Formula: see text] with decidable membership problems, such that [Formula: see text] has an undecidable membership problem. In addition, if [Formula: see text] denotes the pseudovariety of all finite aperiodic semigroups, [Formula: see text] denotes the pseudovariety of all finite groups, and [Formula: see text](E) denotes the pseudovariety of all finite aperiodic semigroups satisfying the finite number of equations E, then it is proved that there exists E such that [Formula: see text](E) has an undecidable membership problem. Note [Formula: see text] equals all semigroups of complexity ≤1. Section 6 is expanded into a joint paper with B. Steinberg, following this paper.

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SCALING OF THE PROTAGONIST IN A. PTUSHKO FILM "NEW GULLIVER" AND THE IDENTITY PROBLEM IN THE SOVIET CHILDREN CINEMA OF THE SECOND HALF OF THE 1930s

RSUH/RGGU Bulletin Series Philosophy Social Studies Art Studies ◽

10.28995/2073-6401-2015-1-73-83 ◽

2015 ◽

pp. 73-83

Author(s):

Nina Yu. Sputnitskaya ◽

◽

Keyword(s):

Identity Problem

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Kant, Marginal Cases, and Moral Standing

10.1093/oso/9780198753858.003.0005 ◽

2018 ◽

Author(s):

Christine M. Korsgaard

Keyword(s):

Life Stage ◽

The Other ◽

Moral Standing ◽

Human Beings ◽

Identity Problem ◽

The Subject ◽

The Dead ◽

Moral Difference

According to the marginal cases argument, there is no property that might justify making a moral difference between human beings and the other animals that is both uniquely and universally human. It is therefore “speciesist” to treat human beings differently just because we are human beings. While not challenging the conclusion, this chapter argues that the marginal cases argument is metaphysically misguided. It ignores the differences between a life stage and a kind, and between lacking a property and having it in a defective form. The chapter then argues for a view of moral standing that attributes it to the subject of a life conceived as an atemporal being, and shows how this view can resolve some familiar puzzles such as how death can be a loss to the person who has died, how we can wrong the dead, the “procreation asymmetry,” and the “non-identity problem.”

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