nLab weakly reductive semigroup




A semigroup is called left/right weakly reductive if it coincides with the semigroup of its left/right translations.


We only define left weakly reductive semigroups, right weakly reductive semigroups are defined similarly.

Let S lS_l be the set of left translations of SS. That is, this is the set of maps x l:SSx_l:S\to S defined by x l(y):=xyx_l(y) := x\cdot y. The semigroup (S l,)(S_l, \circ), where \circ denotes composition of maps, is called the semigroup of left translations of SS.

The map f:xx lf:x\mapsto x_l is then a morphism in the category of semigroups. We call (S,)(S, \cdot) left weakly reductive, if ff is an isomorphism.

Explicitly, and this is where the name comes from, if a,bSa, b\in S, and xa=xbx\cdot a = x\cdot b for all xSx\in S, then a=ba = b.

A weakly reductive semigroup is a semigroup that is both right and left weakly reductive.

In terms of varieties

A left weakly reductive semigroup can be thought of as a class of structures (S,,w,r)(S, \cdot , w, r), where ,w\cdot, w are binary operations, and rr is a ternary operation, satisfying the following axioms for all x,y,zx, y, z: x(yz)=(xy)z,r(x,y,w(x,y)x)=x,r(x,y,w(x,y)y)=y x\cdot (y\cdot z) = (x\cdot y)\cdot z, r(x, y, w(x, y)\cdot x) = x, r(x, y, w(x, y)\cdot y) = y .


Any left monoid, a semigroup with a left identity element, is a left weakly reductive semigroup. In particular, any monoid is weakly reductive.

Any left weakly reductive commutative semigroup is weakly reductive.

A monogenic semigroup that isn’t a group is not weakly reductive.

There exists unique smallest left weakly reductive semigroup which isn’t a left monoid. It can be defined as the idempotent semigroup ({x,y,z},)(\{x, y, z\}, \cdot) such that ab=za\cdot b = z for aba\neq b.


  • A. H. Preston and G. B. Clifford, The algebraic theory of semigroups: Volume I, American Mathematical Society (1961)

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