Green's relations

More generally, the intersection of any L-class with any R-class is either an H-class or the empty set. Green s Theorem states that for any H-class H of a semigroup S either (i) or (ii) H2 = H and H is a subgroup of S. The egg-box for this semigroup would contain infinitely many eggs, but all eggs are in the same box because there is only one D-class.

In a finite semigroup, D and J are the same. There is also a formulation of D in terms of equivalence classes, derived directly from the above definition: Consequently, the D-classes of a semigroup can be seen as unions of L-classes, as unions of R-classes, or as unions of H-classes. John Mackintosh Howie, a prominent semigroup theorist, described this work as so all-pervading that, on encountering a new semigroup, almost the first question one asks is What are the Green relations like? (Howie 2002).

Dually, R is left-compatible: if a R b, then ca R cb. If S is commutative, then L, R and J coincide. The remaining relations are derived from L and R. Six elements of T3 are not in any subgroup. There are essentially two ways of generalising an algebraic theory.

The class Ha is the intersection of La and Ra. The relations are useful for understanding the nature of divisibility in a semigroup; they are also valid for groups, but in this case tell us nothing useful, because groups always have divisibility.

For an element a of S, the relevant ideals are: For elements a and b of S, Green s relations L, R and J are defined by That is, a and b are L-related if they generate the same left ideal; R-related if they generate the same right ideal; and J-related if they generate the same two-sided ideal. For example, L is right-compatible: if a L b and c is another element of S, then ac L bc.

One is to change its definitions so that it covers more or different objects; the other, more subtle way, is to find some desirable outcome of the theory and consider alternative ways of reaching that conclusion. Following the first route, analogous versions of Green s relations have been defined for semirings (Grillet 1970) and rings (Petro 2002). A typical result (Satoh, Yama, and Tokizawa 1994) shows that there are exactly 1,843,120,128 non-equivalent semigroups of order 8, including 221,805 which are commutative; their work is based on a systematic exploration of possible D-classes.

Their intersection is H: This is also an equivalence relation on S. Write (a b c) for the function which sends 1 to a, 2 to b, and 3 to c.

The opposite case, found for example in the bicyclic semigroup, is where each element is in an H-class of its own. (Beware that unit does not mean identity in this context, i.e.

They are also J-classes, because these relations coincide for a finite semigroup. In T3, two functions are L-related if and only if they have the same image. (That is, if two elements of an L-class are in the same R-class, then their images under a bijection will still be in the same R-class.) The dual statement for x L y also holds.

There are also six subgroups of order 2, and three of order 1 (as well as subgroups of these subgroups). in general there are non-identity elements in H1.

Any H-class containing one of these is a (maximal) subgroup. Staying within the world of semigroups, Green s relations can be extended to cover relative ideals, which are subsets that are only ideals with respect to a subsemigroup (Wallace 1963). For the second kind of generalisation, researchers have concentrated on properties of bijections between L- and R- classes.

The question that arises is: how else could there be such bijections? Suppose that Λ and Ρ are semigroups of partial transformations of some semigroup S. Consequently, two functions are D-related if and only if their images are the same size. The elements in bold are the idempotents.

Some, but not all, of the properties associated with the relations in semigroups carry over to these cases. (The join for equivalence relations is normally more difficult to define, but is simplified in this case by the fact that a L c and c R b for some c if and only if a R d and d L b for some d.) As D is the smallest equivalence relation containing both L and R, we know that a D b implies a J b — so J contains D.

Mathematicians today tend to use script letters such as instead, and replace Green s modular arithmetic-style notation with the infix style used here. (By contrast, there are only five groups of order 8.) The full transformation semigroup T3 consists of all functions from the set {1, 2, 3} to itself; there are 27 of these.

Such functions appear in the same column of the table above. In the language of lattices, D is the join of L and R.

For a group, there is only one egg, because all five of Green s relations coincide, and make all group elements equivalent. One example would be to take Λ to be the semigroup of all left translations on S1, restricted to S, and Ρ the corresponding semigroup of restricted right translations. These definitions are due to Clark and Carruth (1980).

There are several choices of partial transformation semigroup that yield the original relations. The full axioms are fairly lengthy to state; informally, the most important requirements are that both Λ and Ρ should contain the identity transformation, and that elements of Λ should commute with elements of Ρ. .

The L-class of a is denoted La (and similarly for the other relations). Green used the lowercase blackletter , and for these relations, and wrote for a L b (and likewise for R and J). In a monoid M, H1 is traditionally called the group of units.

Ordinary letters are used for the equivalence classes. The L and R relations are left-right dual to one another; theorems concerning one can be translated into similar statements about the other. No H-class can contain more than one idempotent, thus H is idempotent separating.

(A semigroup for which all elements are D-related is called bisimple.) It can be shown that within a D-class, all H-classes are the same size. Generalisation of J is not part of this system, as it plays no part in the desired property. We call (Λ, Ρ) a Green s pair.

Clifford and Preston (1961) suggest thinking of this situation in terms of an egg-box: Each row of eggs represents an R-class, and each column an L-class; the eggs themselves are the H-classes. (S1 is S with an identity adjoined if necessary ; if S is not already a monoid, a new element is adjoined and defined to be an identity.) This ensures that principal ideals generated by some semigroup element do indeed contain that element.

These are equivalence relations on S, so each of them yields a partition of S into equivalence classes. In particular, the third D-class is isomorphic to the symmetric group S3.

(Conventionally, arguments are written on the right for Λ, and on the left for Ρ.) Then the L and R relations can be defined by and D and H follow as normal. These bijections are right and left translations, restricted to the appropriate equivalence classes.

They subsume Wallace s work, as well as various other generalised definitions proposed in the mid-1970s. For example, the transformation semigroup T4 contains four D-classes, within which the H-classes have 1, 2, 6, and 24 elements respectively. Recent advances in the combinatorics of semigroups have used Green s relations to help enumerate semigroups with certain properties.

Since T3 contains the identity map, (1 2 3), there is no need to adjoin an identity. The egg-box diagram for T3 has three D-classes. Under certain conditions, it can be shown that if x Ρ = y Ρ, with x ρ1 = y and y ρ2 = x, then the restrictions are mutually inverse bijections.

The unit terminology comes from ring theory.) For example, in the transformation monoid on n elements, Tn, the group of units is the symmetric group Sn. Finally, D is defined by : a D b if and only if there exists a c in S such that a L c and c R b. (In the same way, the ideals of a field are a much less rich environment for study than the ideals of a ring.) Instead of working directly with a semigroup S, we define Green s relations over the monoid S1.

In mathematics, Green s relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. If x R y, then it is always possible to find bijections between Lx and Ly that are R-class-preserving.

An important corollary is that the equivalence class He, where e is an idempotent, is a subgroup of S (its identity is e, and all elements have inverses), and indeed is the largest subgroup of S containing e. Likewise, the functions f and g are R-related if and only if for x and y in {1, 2, 3}; such functions are in the same table row.

The relations are named for James Alexander Green, who introduced them in a paper of 1951.