John Rhodes
580 California St., Suite 400
San Francisco, CA, 94104
Maximal subsemigroups of finite semigroups
1968, Journal of Combinatorial Theory
https://doi.org/10.1016/S0021-9800(68)80001-8
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Abstract
If M is a maximal (proper) subsemigroup of a finite semigroup S, then M contains all but one J-class J(M) of S. When J(M)is non-regular J(M)n M = ~ so M = S --J(M). When J(M) is regular either J(M) c3 M = ~ or M n J(M) has a natural form with respect to the Green-Rees coordinates in J(M). Specifically, there exist an isomorphism j : J(M) ~ ~ dl~ B, G, C) of J(M) ~ with a Rees regular matrix semigroup so that j(M c~ J(M)) = G" x A x B, where G' is a maximal subgroup of G or j(M n J(M)) is the complement of a "rectangle" of ,,~-classes of.Z/~ B, G, C). In the first case, (M c3 J(M)) o is a maximal subsemigroup of J(M) ~ In the second, (M n J(M)) o is maximal in J(M) ~ when j(M n J(M)) = r162176 B, G, C) --(G • A' x B ~) for proper subsets A' and B' of A and B, respectively, but need not be when j(Mn J(M)) = G X A X B' or j(M n J(M)) = G X A' X B. The notation of this paper, with slight variations, follows [1]. ~'~ B, G, C) denotes a Rees matrix semigroup with finite index sets A, B, finite group G and C : B • A ~ G o the structure matrix. If J is a J-class of a semigroup S, we denote by j0 the semigroup J w {0}, 0 r J, with multiplication tjlj~ ,
FAQs
AI
What conditions define a maximal subsemigroup within a finite semigroup?
A maximal subsemigroup M is defined such that if M ⊆ T ⊆ S for some subsemigroup T, then either M = T or T = S. This characterization ensures proper inclusion within S.
How does J-class intersection affect maximal subsemigroup properties?
The study shows that a maximal subsemigroup M must either meet every J-class of S or be a union of J-classes. Specifically, if M intersects, it cannot be contained entirely within any J-class.
What types of structures can form from maximal subsemigroups in finite semigroups?
The findings reveal that J(M) can lead to various structures including null J-classes or a regular Rees matrix semigroup. This variation highlights multiple configurations in finite semigroup composition.
Under what conditions does a maximal subsemigroup provide a union of J-classes?
If the maximal subsemigroup M meets all J-classes of S, it can be shown that M is a union of J-classes. This conclusion stems from the properties of M under the defined intersection.
How does non-regularity in J-class influence maximal subsemigroup behavior?
If J(M) is non-regular, then M must consist of elements from S excluding J(M), indicating a direct relationship between the structure's regularity and its maximal attributes. Consequently, J(M) being non-regular constrains M's elements significantly.
References (3)
- A. H. CLIFFORD AND G. B. PRESTON, Algebraic Theory of Semigroups, (Vol. 1, Math. Surveys No. 7), American Mathematical Society, Providence, R. I., 1962.
- J. A. GREEN, On the Structure of Semigroups, Ann. Math. 54 (1951), 163-172.
- D. REES, On Semigroups, Proc. Cambridge Philos. Soc. 36 (1940), 387-400.