Main Index Algebraic structures Structures with one operation Groupoids
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Special type elements

Let be a groupoid , i.e. a non-empty set with a binary operation defined on it.

If the underlying set is finite then the operation can be given by a table, the so-called multiplication table or Cayley square (table), Thus if has four elements, say, , then the corresponding Cayley table may look as follows

Thus for instance, , or , etc.

A Cayley table of a quasigroup is a Latin square, i.e., if every entry appears in every column and every row exactly once.

An element is called a left unity (or left identity) if for every . Similarly,   is a right unity (or right identity) if for every . A groupoid may have more than one left identify element. For instance, if for all , then  every element of is a  left unity.

If a groupoid has both a left and a right unity, they are necessarily unique and equal. Namely, if is a left unity, and a right unity then . An element which is both a left and a right unity is an unity (or identity) element.

Let be a groupoid with unity . Then element is called left inverse of element if . Analogically, is called right inverse of element . If element is simultaneously left and right inverse of element , i.e. then it is called the inverse of ., An element is said to be regular or invertible if its inverse exists in .  

Example: Let the operation on groupoid is given by the table

The element is the unity element under . Consequently, is its own left and also right inverse. The same is true also for . In addition is also right inverse of . The element has one left inverse , and one right inverse . The element has no right inverse, but it has two left inverses, etc.

If the operation under consideration is multiplication then the unity element is often denoted by (or if the underlying groupoid should be stressed), and the inverse element to element by . If the operation is denoted as addition then unity element is called neutral or zero element denoted or , and the inverse element to is denoted as and called additive inverse or the opposite element. Clearly, for all . However, in many cases it is necessary to distinguish between neutral and zero element, e.g. if the operation on is not necessarily an addition, then   is called a left (right) zero element with respect to   if () for all .  

An element of a groupoid is called a left, or right annihilator if , or , respectively, each . An annihilator is an element which is either left and right annihilator. If the groupoid operation is written additively, then we get the left or right zero element.

Lemma. If a groupoid has a left and a right annihilator then both coincide, and it is the only left and only right annihilator of the groupoid.

The proof is trivial , where is the left and the right annihilator, respectively.

An element is called left cancelable in if implies for all . Similarly, an element is called right cancelable in if implies for all .  If is simultaneously left and right cancelable, it is called cancelable in .  

If we define the left translation by for , then is injective if and only if is left cancelable. Similarly for the right translation .

A groupoid is a groupoid with right cancellation  if  each its element is right cancelable, and a groupoid with left cancellation if  each its element is left cancelable.

An element is called involuted element if , where is the unity element. In this case is its own inverse. Every unity is involutory. Element in the groupoid above is involuted.

An element is called idempotent if .

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