A right congruence p of a semigroup S is called modular if there is an element e of S such that eapa for all a in S. The element e is called a left identity for p. A similar definition is made for modular left congruences. A two-sided congruence is called modular if it is modular both as a right and left congruence and hence has a two-sided identity. We define RT[R¡, P¡] to be the intersection of all the maximal modular right [left, two-sided] congruences. In each case the intersection is a two-sided congruence. We shall refer to them as the r[I-, t-] radical of S. A semigroup ,S" is said to be x-semisimple if the x-radical is the identity relation i on S. A semigroup modulo its x-radical is x-semisimple. The maximal modular two-sided congruences of a finite semigroup can be described by two mutually exclusive types. This classification is then used to prove that a r-semisimple finite semigroup is f-semisimple if and only if it is a semilattice Y of groups Ga, a e Y such that the structure homomorphisms <paß :Ga^Gs (a> ß) [1, p. 128] are one-to-one and the group G0 (0 the minimal element of Y) is tsemisimple, i.e., a group whose Frattini subgroup is trivial. The maximal modular right congruences of a finite semigroup can be classified into three mutually disjoint classes. This classification is then used to prove that the kernel of a finite r-semisimple semigroup is right simple, i.e., consists of a single minimal right ideal. A similar approach, i.e., via congruence relations, to the structure of semigroups has been initiated by Hoehnke [6]. In his paper Hoehnke gives several definitions of radicals; rad S, (rad)~ S, rad° S, etc., which can be related to intersections of modular congruences. (The concept of a modular congruence was also introduced in [7] and [9].) It should be pointed out that for a semigroup S, (rad)"=r-rad and if S is finite rad = r-rad. The author expresses his gratitude to the referee for his contributions to the revision of this paper, in particular, for the shortened versions of the proofs of Theorems 11, 13, and 28.
[1]
Robert H. Oehmke.
On the Structures of an Automaton and Its Input Semigroup
,
1963,
JACM.
[2]
D. Rees,et al.
On semi-groups
,
1940,
Mathematical Proceedings of the Cambridge Philosophical Society.
[3]
Paul Dubreil,et al.
Contribution à la théorie des demi-groupes. III
,
1953
.
[4]
M. Hall.
The Theory Of Groups
,
1959
.
[5]
R. Dean,et al.
Idempotent semigroups with distributive right congruence lattices
,
1964
.
[6]
A. Clifford,et al.
The algebraic theory of semigroups
,
1964
.
[7]
Representation of a Semigroup by Transformations Acting Transitively on a Set
,
1961
.
[8]
W. D. Munn,et al.
The Algebraic Theory of Semigroups, Vol. I
,
1964,
The Mathematical Gazette.