In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five distinct nonisomorphic semigroups having two elements:
Contents
- Determination of semigroups with two elements
- The two element semigroup 01
- The two element semigroup Z22
- Semigroups of order 3
- Finite semigroups of higher orders
- References
The semigroups LO2 and RO2 are antiisomorphic. O2, ({0,1}, ∧) and (Z2, +2) are commutative, LO2 and RO2 are noncommutative. LO2, RO2 and ({0,1}, ∧) are bands and also inverse semigroups.
Determination of semigroups with two elements
Choosing the set A = { 1, 2 } as the underlying set having two elements, sixteen binary operations can be defined in A. These operations are shown in the table below. In the table, a matrix of the form
indicates a binary operation on A having the following Cayley table.
In this table:
The two-element semigroup ({0,1}, ∧)
The Cayley table for the semigroup ({0,1},
This is the simplest non-trivial example of a semigroup that is not a group. This semigroup has an identity element, 1, making it a monoid. It is also commutative. It is not a group because the element 0 does not have an inverse, and is not even a cancellative semigroup because we cannot cancel the 0 in the equation 1·0 = 0·0.
This semigroup arises in various contexts. For instance, if we choose 1 to be the truth value "true" and 0 to be the truth value "false" and the operation to be the logical connective "and", we obtain this semigroup in logic. It is isomorphic to the monoid {0,1} under multiplication. It is also isomorphic to the semigroup
under matrix multiplication.
The two-element semigroup (Z2,+2)
The Cayley table for the semigroup (Z2,+2) is given below:
This group is isomorphic to the cyclic group Z2 and the symmetric group S2.
Semigroups of order 3
Let A be the three-element set {1, 2, 3}. Altogether, a total of 39 = 19683 different binary operations can be defined on A. 113 of the 19683 binary operations determine 24 nonisomorphic semigroups, or 18 non-equivalent semigroups (with equivalence being isomorphism or anti-isomorphism). With the exception of the group with three elements, each of these has one (or more) of the above two-element semigroups as subsemigroups. For example, the set {−1,0,1} under multiplication is a semigroup of order 3, and contains both {0,1} and {−1,1} as subsemigroups.
Finite semigroups of higher orders
Algorithms and computer programs have been developed for determining nonisomorphic finite semigroups of a given order. These have been applied to determine the nonisomorphic semigroups of small order. The number of nonisomorphic semigroups with n elements, for n a nonnegative integer, is listed under A027851 in the On-Line Encyclopedia of Integer Sequences. A001423 lists the number of non-equivalent semigroups, and A023814 the number of associative binary operations, out of a total of nn2, determining a semigroup.