| dc.description.abstract |
Since the work of Marczewski [10], many authors have studied the rank properties in
the context of general algebras (cf. [1–3,9,11,12,14]). The concept of rank for general
algebras is analogous to the concept of dimension in linear algebra. The dimension of
a vector space is the maximum cardinality of an independent subset, or equivalently,
it is the minimum cardinality of a generating set of the vector space. A subset U of a
semigroup is said to be independent if every element ofU is not in the subsemigroup
generated by the remaining elements of U, i.e.,
∀a ∈ U, a /∈ U \ {a} .
It can be observed that the minimum size of a generating set need not be equal to the
maximum size of an independent set in a semigroup. Accordingly, Howie and Ribeiro
have considered various concepts of ranks for a finite semigroup (cf. [5,6]).
1. r1( ) = max{k: every subset U of cardinality k in is independent}.
2. r2( ) = min{|U| : U ⊆ , U = }.
3. r3( ) = max{|U| : U ⊆ , U = ,U is independent}.
4. r4( ) = max{|U| : U ⊆ ,U is independent}. |
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