# examples of algebraic systems

Selected examples of algebraic systems are specified below.

1. 1.
2. 2.

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

3. 3.
4. 4.

A monoid is an algebra of type $\langle 2,0\rangle$. However, not every algebra of type $\langle 2,0\rangle$ is a monoid.

5. 5.

A group is an algebraic system of type $\langle 2,1,0\rangle$, 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 $\langle 2,2,1,0,0\rangle$, 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.
8. 8.
9. 9.
10. 10.

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

11. 11.

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

12. 12.

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

13. 13.

A left module over a ring $R$ is an algebraic system. Its type is $\langle 2,1,(1)_{r\in R},0\rangle$, 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 $r\in R$, and finally $0$ the (arity) of additive identity.

14. 14.

The set $\overline{V}$ 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 $\overline{V}$ may be associated with a finitary operation on $\overline{V}$. In classical propositional logic  , the algebraic system may be of type $\langle 1,2\rangle$, if we consider $\neg$ and $\vee$ as the only logical connectives; or it may be of type $\langle 1,2,2,2,2\rangle$, if the full set $\{\neg,\vee,\wedge,\to,\leftrightarrow\}$ is used.

Below are some non-examples of algebraic systems:

1. 1.
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.

## References

• 1 G. Grätzer: , 2nd Edition, Springer, New York (1978).
• 2 P. Jipsen: Mathematical Structures: Homepage
Title examples of algebraic systems ExamplesOfAlgebraicSystems 2013-03-22 18:40:11 2013-03-22 18:40:11 CWoo (3771) CWoo (3771) 16 CWoo (3771) Example msc 08A05 msc 03E99 msc 08A62