Definition:Properties of Algebraic Structures of Two Operations
The purpose of this page is to gather into one place the various types of algebraic structure of two binary operations: $+, *$, and to classify them according to the properties they hold. The properties are defined by the following key: {| border="1" | $A0$ | Closure of $+$ |- | $A1$ | Associativity of $+$ |- | $A2$ | Identity Element of $+$ |- | $A3$ | Inverses of $+$ |- | $A4$ | Commutativity of $+$ |- | $M0$ | Closure of $+$ |- | $M1$ | Associativity of $*$ |- | $M2$ | Identity Element of $*$ |- | $M3$ | Inverses of $*$ |- | $M4$ | Commutativity of $*$ |- | $D$ | Distributivity of $*$ over $+$ |- | $NZD$ | No (Proper) Zero Divisors |- | $AC2$ | Characteristic 2 for $+$ |- | $AZ$ | Zero Element of $+$ |- | $D'$ | Distributivity of $+$ over $*$ |- | $MI$ | Idempotence of $*$ |- | $MZ$ | Zero Element of $*$ |- |} {| border="1" !  ! $A0$ ! $A1$ ! $A2$ ! $A3$ ! $A4$ ! $M0$ ! $M1$ ! $M2$ ! $M3$ ! $M4$ ! $D$  ! $NZD$ ! $AC2$  ! $AZ$ ! $MI$ ! $MZ$ ! $D'$ |- | Semiring | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  |  |  | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | |  | align="center"| $\checkmark$ |  |  | |  | |  |- | Additive Semiring | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  |  | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | |  | align="center"| $\checkmark$ |  |  | |  | |  |- | Rig | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | |  | align="center"| $\checkmark$ |  |  | |  | align="center"| $\checkmark$ |  |- | Ring without associativity | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | | |  | align="center"| $\checkmark$ |  |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Ring | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | |  | align="center"| $\checkmark$ |  |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Ring with Unity | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | |  | align="center"| $\checkmark$ |  |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Commutative Ring with Unity | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Integral Domain | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Division Ring | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Field | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  | |  | align="center"| $(\checkmark)$ (see here) |  |- | Idempotent Ring | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | |  |  | align="center"| $\checkmark$ | | align="center"| $\checkmark$ |  | align="center"| $\checkmark$ |  |  |- | Boolean Ring | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |  |  | align="center"| $\checkmark$ | | align="center"| $\checkmark$ |  | align="center"| $\checkmark$ |  |  |- | Boolean Algebra | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | | | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ | align="center"| $\checkmark$ |- |} A '''checkmark''' in brackets: $(\checkmark)$ denotes that the property indicated can be derived from the others.