examples of algebraic systems

Selected examples of algebraic systems are specified below.

1. A set is an algebra where $\tau =\mathrm{\varnothing }$.

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

3. An algebra of type $⟨2⟩$ is called a groupoid. Another algebra of this type is a semigroup.

4. A monoid is an algebra of type $⟨2,0⟩$. However, not every algebra of type $⟨2,0⟩$ is a monoid.

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. 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. A lattice is an algebraic system of type $⟨2,2⟩$. The two binary operations are meet and join.

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

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

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 identity, is an algebraic system of type $⟨0,q⟩$, where $q$ is the type of a quasigroup.

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

13. A left module over a ring $R$ is an algebraic system. Its type is $⟨2,1,{(1)}_{r\in R},0⟩$, 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 $r\in R$, 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 $⟨1,2⟩$, if we consider $\mathrm{\neg }$ and $\vee$ as the only logical connectives; or it may be of type $⟨1,2,2,2,2⟩$, if the full set $\{\mathrm{\neg },\vee ,\wedge ,\to ,\leftrightarrow \}$ 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.