examples of algebraic systems

Selected examples of algebraic systems are specified below.

  1. 1.

    A set is an algebraMathworldPlanetmathPlanetmath where τ=.

  2. 2.

    A pointed set is an algebra of type 0, where 0 corresponds to the designated element in the set.

  3. 3.

    An algebra of type 2 is called a groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Another algebra of this type is a semigroupPlanetmathPlanetmath.

  4. 4.

    A monoid is an algebra of type 2,0. However, not every algebra of type 2,0 is a monoid.

  5. 5.

    A group is an algebraic system of type 2,1,0, where 2 corresponds to the arity of the multiplication, 1 the multiplicative inverse, and 0 the multiplicative identity.

  6. 6.

    A ring is an algebraic system of type 2,2,1,0,0, where the two 2’s represent the arities of addition and multiplication, 1 the additive inverse, and 0’s the additive and multiplicative identities.

  7. 7.

    A latticeMathworldPlanetmath is an algebraic system of type 2,2. The two binary operationsMathworldPlanetmath are meet and join.

  8. 8.

    A bounded latticeMathworldPlanetmath is an algebraic system of type 2,2,0,0. Besides the meet and join operationsMathworldPlanetmath, it has two constants, its top 1 and bottom 0.

  9. 9.

    A uniquely complemented latticeMathworldPlanetmath is an algebraic system of the type 2,2,1,0,0. In addition to having the operations of a bounded lattice, there is a unary operator taking each element to its unique complementPlanetmathPlanetmath. Note that it has the same type as the type of a group.

  10. 10.

    A quandle is an algebraic system of type 2,2,. It has the same type as a lattice.

  11. 11.

    A quasigroup may be thought of as a algebraic system of type 2, that of a groupoid, or 2,2,2, depending on the definition used. A loop, as a quasigroup with an identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, is an algebraic system of type 0,q, where q is the type of a quasigroup.

  12. 12.

    An n-group (http://planetmath.org/PolyadicSemigroup) is an algebraic system of type n.

  13. 13.

    A left module over a ring R is an algebraic system. Its type is 2,1,(1)rR,0, where 2 is the arity of addition, the first 1 the additive inverse, and the rest of the 1’s represent the arity of left scalar multiplication by r, for each rR, and finally 0 the (arity) of additive identity.

  14. 14.

    The set V¯ of all well-formed formulas over a set V of propositional variables can be thought of as an algebraic system, as each of the logical connectives as an operation on V¯ may be associated with a finitary operation on V¯. In classical propositional logicPlanetmathPlanetmath, the algebraic system may be of type 1,2, if we consider ¬ and as the only logical connectives; or it may be of type 1,2,2,2,2, if the full set {¬,,,,} is used.

Below are some non-examples of algebraic systems:

  1. 1.

    A complete latticeMathworldPlanetmath is not, in general, an algebraic system because the arbitrary meet and join operations are not finitary.

  2. 2.

    A field is not an algebraic system, since, in addition to the five operations of a ring, there is the multiplicative inverse operation, which is not defined for 0.

  3. 3.

    A small category may be defined as a set with one partial binary operation on it. Unless the category has only one object (so that the operation is everywhere defined), it is in general not an algebraic system.


  • 1 G. Grätzer: Universal AlgebraMathworldPlanetmath, 2nd Edition, Springer, New York (1978).
  • 2 P. Jipsen: http://math.chapman.edu/cgi-bin/structuresMathworldPlanetmath?HomePageMathematical Structures: Homepage
Title examples of algebraic systems
Canonical name ExamplesOfAlgebraicSystems
Date of creation 2013-03-22 18:40:11
Last modified on 2013-03-22 18:40:11
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 16
Author CWoo (3771)
Entry type Example
Classification msc 08A05
Classification msc 03E99
Classification msc 08A62