Concept
Trivial semigroup
Related concepts (4)
Semigroup with two elements
In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements: O2, the null semigroup of order two, LO2, the left zero semigroup of order two, RO2, the right zero semigroup of order two, ({0,1}, ∧) (where "∧" is the logical connective "and"), or equivalently the set {0,1} under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order two, also a monoid, and ultimately the two-element Boolean algebra, (Z2, +2) (where Z2 = {0,1} and "+2" is "addition modulo 2"), or equivalently ({0,1}, ⊕) (where "⊕" is the logical connective "xor"), or equivalently the set {−1,1} under multiplication: the only group of order two.
Empty semigroup
In mathematics, a semigroup with no elements (the empty semigroup) is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation. However not all authors insist on the underlying set of a semigroup being non-empty. One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function from S × S to S.
Semigroup with three elements
In abstract algebra, a semigroup with three elements is an object consisting of three elements and an associative operation defined on them. The basic example would be the three integers 0, 1, and −1, together with the operation of multiplication. Multiplication of integers is associative, and the product of any two of these three integers is again one of these three integers.
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): x·y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x, y). Associativity is formally expressed as that (x·y)·z = x·(y·z) for all x, y and z in the semigroup.