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# examples of algebraic systems

Selected examples of algebraic systems are specified below.

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

3. 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. 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. 7. A lattice is an algebraic system of type $\langle 2,2\rangle$. The two binary operations are meet and join.

8. A bounded lattice is an algebraic system of type $\langle 2,2,0,0\rangle$. Besides the meet and join operations, it has two constants, its top $1$ and bottom $0$.

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

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

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. An $n$-group is an algebraic system of type $\langle n\rangle$.

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

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. 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: Universal Algebra, 2nd Edition, Springer, New York (1978).
- 2 P. Jipsen: Mathematical Structures: Homepage

## Mathematics Subject Classification

08A05*no label found*03E99

*no label found*08A62

*no label found*

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