{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "KRM89R0T28"
    },
    {
      "field": "uses",
      "target": "KRM89R0T28"
    }
  ],
  "body": "\n## Problem\n\nA finite algebra $A$ of finite signature is given by the finite tables of its operations.\nIt is [[finitely based|YGYZAQ12MZ]] if it has a finite [[equational basis|9Q008GMZSW]].\nThat is, some finite set of [[identities|PKD440G2XW]] true in $A$ implies every identity true in $A$.\nTarski's finite basis problem asks for an algorithm, that is, a [[computable function|KZSB5E471K]], with the following task.\nGiven a finite algebra of finite signature, it decides whether the algebra is finitely based.\n\n## Remarks\n\nThe problem was solved negatively by Ralph McKenzie in 1996: no such algorithm exists.\nEarlier work showed that the answer to \"is every finite algebra finitely based?\" is no.\nLyndon showed that every two-element algebra of finite signature is finitely based, and he also gave a seven-element algebra that is not.\nMany classes of finite algebras are known to consist of finitely based algebras, among them finite groups, by the [[Oates–Powell theorem|5RFPT46DXX]], and finite lattices.\n[[Baker's finite basis theorem|35G4HQG2FN]] covers all finite algebras of finite signature that generate a congruence-distributive variety.\nThe decision problem restricted to finite semigroups is still open.\n\n## Explanations\n\n### L1 human\n\nThe identities true in a finite algebra form an infinite list.\nSometimes finitely many of them imply all the others, and sometimes they do not.\nTarski asked whether a computer could always tell which case happens, just by looking at the tables of the operations.\nMcKenzie proved that no computer program can do this for all finite algebras.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "YGYZAQ12MZ"
    ],
    "history": {
      "posed_by": "Alfred Tarski"
    },
    "kind": "problem",
    "msc": [
      "08B05",
      "03B25",
      "03C05"
    ],
    "name": "Tarski's finite basis problem",
    "sources": [
      {
        "url": "https://doi.org/10.1142/S0218196796000040"
      },
      {
        "url": "https://en.wikipedia.org/wiki/Variety_(universal_algebra)"
      }
    ],
    "tag": "AB22FWBQ0M",
    "uses": [
      "YGYZAQ12MZ",
      "9Q008GMZSW",
      "PKD440G2XW",
      "KZSB5E471K",
      "5RFPT46DXX",
      "35G4HQG2FN"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/AB22FWBQ0M/",
  "links": [
    {
      "field": "about",
      "target": "YGYZAQ12MZ"
    },
    {
      "field": "uses",
      "target": "YGYZAQ12MZ"
    },
    {
      "field": "uses",
      "target": "9Q008GMZSW"
    },
    {
      "field": "uses",
      "target": "PKD440G2XW"
    },
    {
      "field": "uses",
      "target": "KZSB5E471K"
    },
    {
      "field": "uses",
      "target": "5RFPT46DXX"
    },
    {
      "field": "uses",
      "target": "35G4HQG2FN"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AB22FWBQ0M.md",
  "msc": [
    "08B05",
    "03B25",
    "03C05"
  ],
  "name": "Tarski's finite basis problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AB2/AB22FWBQ0M.md",
  "status": null,
  "tag": "AB22FWBQ0M",
  "uri": "m2://AB22FWBQ0M",
  "url": "https://tamnd.github.io/math2/wiki/tarskis-finite-basis-problem/",
  "writing": {
    "tag": "AB22FWBQ0M",
    "kind": "problem",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}