{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *De Morgan algebra* is a structure $(A, \\vee, \\wedge, 0, 1, \\sim)$ such that:\n\n1. $(A, \\vee, \\wedge, 0, 1)$ is a [[bounded lattice|72V1Z82M5F]] that is a [[distributive lattice|0HM0HQCK6N]].\n2. ${\\sim} \\colon A \\to A$ satisfies ${\\sim}{\\sim} a = a$ for every $a \\in A$.\n3. ${\\sim}(a \\vee b) = {\\sim} a \\wedge {\\sim} b$ for all $a, b \\in A$.\n\n## Remarks\n\nFrom clauses 2 and 3 one gets ${\\sim}(a \\wedge b) = {\\sim} a \\vee {\\sim} b$, ${\\sim} 0 = 1$ and ${\\sim} 1 = 0$.\nSo ${\\sim}$ is an order-reversing bijection of $A$ onto itself.\n\nExample: the interval $[0, 1]$ with $a \\vee b = \\max(a, b)$, $a \\wedge b = \\min(a, b)$ and ${\\sim} a = 1 - a$.\nA chain is a distributive lattice, $1 - (1 - a) = a$, and $1 - \\max(a, b) = \\min(1 - a, 1 - b)$.\nIt is not a Boolean algebra, since $\\tfrac12 \\wedge {\\sim} \\tfrac12 = \\tfrac12 \\ne 0$.\n\nNon-example: the chain $0 \u003c m \u003c 1$ with ${\\sim}$ the [[pseudocomplement|WMR73J0S26]], so ${\\sim} 0 = 1$, ${\\sim} m = 0$, ${\\sim} 1 = 0$.\nClause 2 fails, since ${\\sim}{\\sim} m = 1 \\ne m$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "5Q53XWV0QN",
    "kind": "definition",
    "msc": [
      "06D30"
    ],
    "name": "De Morgan algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/De_Morgan_algebra"
      }
    ],
    "tag": "CDGBK86D23",
    "uses": [
      "72V1Z82M5F",
      "0HM0HQCK6N"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/CDGBK86D23/",
  "links": [
    {
      "field": "defines",
      "target": "5Q53XWV0QN"
    },
    {
      "field": "uses",
      "target": "72V1Z82M5F"
    },
    {
      "field": "uses",
      "target": "0HM0HQCK6N"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CDGBK86D23.md",
  "msc": [
    "06D30"
  ],
  "name": "De Morgan algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CDG/CDGBK86D23.md",
  "status": null,
  "tag": "CDGBK86D23",
  "uri": "m2://CDGBK86D23",
  "url": "https://tamnd.github.io/math2/wiki/de-morgan-algebra-definition/",
  "writing": {
    "tag": "CDGBK86D23",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}