Main Index
Algebraic structures
Structures with one operation
Subject Index
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Main Index
Algebraic structures
Structures with one operation
Subject Index
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A set may be regarded as a trivial algebraic structure with no operation defined. The simplest structures of some interest is a set endowed with one operation.
Nullary operation leads to a pointed set: this is a set with a distinguished element.
Unary system is a set 
with unary operation, equivalently we have a function defined on 
.
Example: Peano axiom system is a pointed unary system. The distinguished element is 
, and the unary operation is the function 
.
The simplest algebraic structures is a set endowed with one binary operation.
A groupoid 
(or magma) is a non-empty set 
together with a binary operation 
. The groupoid is closed under the operation. As a operation signs other symbols are also used, e.g. 
or 
etc, and the operation is called groupoid operation. To stress the operation also the notation 
is used for the groupoid endowed with binary operation 
.
A subset 
is called subgrupoid of 
if 
is also a grupoid with respect to 
.
Quasigroup is a groupoid 
in which every element is invertible with respect to 
.
Loop is a quasigroup with an identity element.
Semigroup is an groupoid 
with 
associative, i.e. a set 
on which an associative binary operation has been defined.
Monoid is a semigroup with an identity element.
Group is a monoid in which every element is invertible, or equivalently, an associative loop.
Cite this web-page as:
Štefan Porubský: Algebraic Structures With One Operation.