Terminology |
Defining property |
Variety of finite semigroup |
Reference(s) |
Finite semigroup |
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Empty semigroup |
- S =

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No |
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Trivial semigroup |
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Monoid |
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No |
Gril p. 3 |
Band (Idempotent semigroup) |
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C&P p. 4 |
Rectangular band |
- A band such that abca = acba
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Fennemore |
Semilattice |
A commutative band, that is:
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Commutative semigroup |
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C&P p. 3 |
Archimedean commutative semigroup |
- ab = ba
- There exists x and k such that ak = xb.
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C&P p. 131 |
Nowhere commutative semigroup |
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C&P p. 26 |
Left weakly commutative |
- There exist x and k such that (ab)k = bx.
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Nagy p. 59 |
Right weakly commutative |
- There exist x and k such that (ab)k = xa.
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Nagy p. 59 |
Weakly commutative |
Left and right weakly commutative. That is:
- There exist x and j such that (ab)j = bx.
- There exist y and k such that (ab)k = ya.
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Nagy p. 59 |
Conditionally commutative semigroup |
- If ab = ba then axb = bxa for all x.
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Nagy p. 77 |
R-commutative semigroup |
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Nagy p. 69–71 |
RC-commutative semigroup |
- R-commutative and conditionally commutative
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Nagy p. 93–107 |
L-commutative semigroup |
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Nagy p. 69–71 |
LC-commutative semigroup |
- L-commutative and conditionally commutative
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Nagy p. 93–107 |
H-commutative semigroup |
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Nagy p. 69–71 |
Quasi-commutative semigroup |
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Nagy p. 109 |
Right commutative semigroup |
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Nagy p. 137 |
Left commutative semigroup |
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Nagy p. 137 |
Externally commutative semigroup |
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Nagy p. 175 |
Medial semigroup |
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Nagy p. 119 |
E-k semigroup (k fixed) |
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Nagy p. 183 |
Exponential semigroup |
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Nagy p. 183 |
WE-k semigroup (k fixed) |
- There is a positive integer j depending on the couple (a,b) such that (ab)k+j = akbk (ab)j = (ab)jakbk
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Nagy p. 199 |
Weakly exponential semigroup |
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Nagy p. 215 |
Right cancellative semigroup |
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C&P p. 3 |
Left cancellative semigroup |
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C&P p. 3 |
Cancellative semigroup |
Left and right cancellative semigroup, that is
- ab = ac ⇒ b = c
- ba = ca ⇒ b = c
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C&P p. 3 |
''E''-inversive semigroup (E-dense semigroup) |
- There exists x such that ax ∈ E.
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C&P p. 98 |
Regular semigroup |
- There exists x such that axa =a.
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C&P p. 26 |
Regular band |
- A band such that abaca = abca
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Fennemore |
Intra-regular semigroup |
- There exist x and y such that xa2y = a.
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C&P p. 121 |
Left regular semigroup |
- There exists x such that xa2 = a.
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C&P p. 121 |
Left-regular band |
- A band such that aba = ab
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Fennemore |
Right regular semigroup |
- There exists x such that a2x = a.
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C&P p. 121 |
Right-regular band |
- A band such that aba = ba
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Fennemore |
Completely regular semigroup |
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Gril p. 75 |
(inverse) Clifford semigroup |
- A regular semigroup in which all idempotents are central.
- Equivalently, for finite semigroup:

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Petrich p. 65 |
k-regular semigroup (k fixed) |
- There exists x such that akxak = ak.
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Hari |
Eventually regular semigroup (π-regular semigroup, Quasi regular semigroup) |
- There exists k and x (depending on a) such that akxak = ak.
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Edwa Shum Higg p. 49 |
Quasi-periodic semigroup, epigroup, group-bound semigroup, completely (or strongly) π-regular semigroup, and many other; see Kela for a list) |
- There exists k (depending on a) such that ak belongs to a subgroup of S
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Kela Gril p. 110 Higg p. 4 |
Primitive semigroup |
- If 0 ≠ e and f = ef = fe then e = f.
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C&P p. 26 |
Unit regular semigroup |
- There exists u in G such that aua = a.
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Tvm |
Strongly unit regular semigroup |
- There exists u in G such that aua = a.
- e D f ⇒ f = v−1ev for some v in G.
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Tvm |
Orthodox semigroup |
- There exists x such that axa = a.
- E is a subsemigroup of S.
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Gril p. 57 Howi p. 226 |
Inverse semigroup |
- There exists unique x such that axa = a and xax = x.
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C&P p. 28 |
Left inverse semigroup (R-unipotent) |
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Gril p. 382 |
Right inverse semigroup (L-unipotent) |
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Gril p. 382 |
Locally inverse semigroup (Pseudoinverse semigroup) |
- There exists x such that axa = a.
- E is a pseudosemilattice.
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Gril p. 352 |
M-inversive semigroup |
- There exist x and y such that baxc = bc and byac = bc.
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C&P p. 98 |
Abundant semigroup |
- The classes L*a and R*a, where a L* b if ac = ad ⇔ bc = bd and a R* b if ca = da ⇔ cb = db, contain idempotents.
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Chen |
Rpp-semigroup (Right principal projective semigroup) |
- The class L*a, where a L* b if ac = ad ⇔ bc = bd, contains at least one idempotent.
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Shum |
Lpp-semigroup (Left principal projective semigroup) |
- The class R*a, where a R* b if ca = da ⇔ cb = db, contains at least one idempotent.
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Shum |
Null semigroup (Zero semigroup) |
- 0 ∈ S
- ab = 0
- Equivalently ab = cd
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C&P p. 4 |
Left zero semigroup |
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C&P p. 4 |
Left zero band |
A left zero semigroup which is a band. That is:
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Left group |
- A semigroup which is left simple and right cancellative.
- The direct product of a left zero semigroup and an abelian group.
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C&P p. 37, 38 |
Right zero semigroup |
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C&P p. 4 |
Right zero band |
A right zero semigroup which is a band. That is:
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Fennemore |
Right group |
- A semigroup which is right simple and left cancellative.
- The direct product of a right zero semigroup and a group.
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C&P p. 37, 38 |
Right abelian group |
- A right simple and conditionally commutative semigroup.
- The direct product of a right zero semigroup and an abelian group.
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Nagy p. 87 |
Unipotent semigroup |
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C&P p. 21 |
Left reductive semigroup |
- If xa = xb for all x then a = b.
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C&P p. 9 |
Right reductive semigroup |
- If ax = bx for all x then a = b.
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C&P p. 4 |
Reductive semigroup |
- If xa = xb for all x then a = b.
- If ax = bx for all x then a = b.
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C&P p. 4 |
Separative semigroup |
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C&P p. 130–131 |
Reversible semigroup |
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C&P p. 34 |
Right reversible semigroup |
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C&P p. 34 |
Left reversible semigroup |
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C&P p. 34 |
Aperiodic semigroup
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- There exists k (depending on a) such that ak = ak+1
- Equivalently, for finite semigroup: for each a,
.
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ω-semigroup |
- E is countable descending chain under the order a ≤H b
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Gril p. 233–238 |
Left Clifford semigroup (LC-semigroup) |
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Shum |
Right Clifford semigroup (RC-semigroup) |
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Shum |
Orthogroup |
- Ha is a group.
- E is a subsemigroup of S
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Shum |
Complete commutative semigroup |
- ab = ba
- ak is in a subgroup of S for some k.
- Every nonempty subset of E has an infimum.
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Gril p. 110 |
Nilsemigroup (Nilpotent semigroup) |
- 0 ∈ S
- ak = 0 for some integer k which depends on a.
- Equivalently, for finite semigroup: for each element x and y,
.
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Elementary semigroup |
- ab = ba
- S is of the form G ∪ N where
- G is a group, and 1 ∈ G
- N is an ideal, a nilsemigroup, and 0 ∈ N
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Gril p. 111 |
E-unitary semigroup |
- There exists unique x such that axa = a and xax = x.
- ea = e ⇒ a ∈ E
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Gril p. 245 |
Finitely presented semigroup |
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Gril p. 134 |
Fundamental semigroup |
- Equality on S is the only congruence contained in H.
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Gril p. 88 |
Idempotent generated semigroup |
- S is equal to the semigroup generated by E.
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Gril p. 328 |
Locally finite semigroup |
- Every finitely generated subsemigroup of S is finite.
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Gril p. 161 |
N-semigroup |
- ab = ba
- There exists x and a positive integer n such that a = xbn.
- ax = ay ⇒ x = y
- xa = ya ⇒ x = y
- E = Ø
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Gril p. 100 |
L-unipotent semigroup (Right inverse semigroup) |
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Gril p. 362 |
R-unipotent semigroup (Left inverse semigroup) |
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Gril p. 362 |
Left simple semigroup |
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Gril p. 57 |
Right simple semigroup |
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Gril p. 57 |
Subelementary semigroup |
- ab = ba
- S = C ∪ N where C is a cancellative semigroup, N is a nilsemigroup or a one-element semigroup.
- N is ideal of S.
- Zero of N is 0 of S.
- For x, y in S and c in C, cx = cy implies that x = y.
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Gril p. 134 |
Symmetric semigroup (Full transformation semigroup) |
- Set of all mappings of X into itself with composition of mappings as binary operation.
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C&P p. 2 |
Weakly reductive semigroup |
- If xz = yz and zx = zy for all z in S then x = y.
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C&P p. 11 |
Right unambiguous semigroup |
- If x, y ≥R z then x ≥R y or y ≥R x.
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Gril p. 170 |
Left unambiguous semigroup |
- If x, y ≥L z then x ≥L y or y ≥L x.
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Gril p. 170 |
Unambiguous semigroup |
- If x, y ≥R z then x ≥R y or y ≥R x.
- If x, y ≥L z then x ≥L y or y ≥L x.
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Gril p. 170 |
Left 0-unambiguous |
- 0∈ S
- 0 ≠ x ≤L y, z ⇒ y ≤L z or z ≤L y
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Gril p. 178 |
Right 0-unambiguous |
- 0∈ S
- 0 ≠ x ≤R y, z ⇒ y ≤L z or z ≤R y
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Gril p. 178 |
0-unambiguous semigroup |
- 0∈ S
- 0 ≠ x ≤L y, z ⇒ y ≤L z or z ≤L y
- 0 ≠ x ≤R y, z ⇒ y ≤L z or z ≤R y
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Gril p. 178 |
Left Putcha semigroup |
- a ∈ bS1 ⇒ an ∈ b2S1 for some n.
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Nagy p. 35 |
Right Putcha semigroup |
- a ∈ S1b ⇒ an ∈ S1b2 for some n.
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Nagy p. 35 |
Putcha semigroup |
- a ∈ S1b S1 ⇒ an ∈ S1b2S1 for some positive integer n
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Nagy p. 35 |
Bisimple semigroup (D-simple semigroup) |
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C&P p. 49 |
0-bisimple semigroup |
- 0 ∈ S
- S - {0} is a D-class of S.
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C&P p. 76 |
Completely simple semigroup |
- There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A.
- There exists h in E such that whenever hf = f and fh = f we have h = f.
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C&P p. 76 |
Completely 0-simple semigroup |
- 0 ∈ S
- S2 ≠ 0
- If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0 or A = S.
- There exists non-zero h in E such that whenever hf = f, fh = f and f ≠ 0 we have h = f.
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C&P p. 76 |
D-simple semigroup (Bisimple semigroup) |
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C&P p. 49 |
Semisimple semigroup |
- Let J(a) = S1aS1, I(a) = J(a) − Ja. Each Rees factor semigroup J(a)/I(a) is 0-simple or simple.
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C&P p. 71–75 |
: Simple semigroup |
- Ja = S. (There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A.),
- equivalently, for finite semigroup:
and .
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0-simple semigroup |
- 0 ∈ S
- S2 ≠ 0
- If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0.
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C&P p. 67 |
Left 0-simple semigroup |
- 0 ∈ S
- S2 ≠ 0
- If A ⊆ S is such that SA ⊆ A then A = 0.
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C&P p. 67 |
Right 0-simple semigroup |
- 0 ∈ S
- S2 ≠ 0
- If A ⊆ S is such that AS ⊆ A then A = 0.
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C&P p. 67 |
Cyclic semigroup (Monogenic semigroup) |
- S = { w, w2, w3, ... } for some w in S
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C&P p. 19 |
Periodic semigroup |
- { a, a2, a3, ... } is a finite set.
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C&P p. 20 |
Bicyclic semigroup |
- 1 ∈ S
- S admits the presentation
.
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C&P p. 43–46 |
Full transformation semigroup TX (Symmetric semigroup) |
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C&P p. 2 |
Rectangular band |
- A band such that aba = a
- Equivalently abc = ac
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Fennemore |
Rectangular semigroup |
- Whenever three of ax, ay, bx, by are equal, all four are equal.
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C&P p. 97 |
Symmetric inverse semigroup IX |
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C&P p. 29 |
Brandt semigroup |
- 0 ∈ S
- ( ac = bc ≠ 0 or ca = cb ≠ 0 ) ⇒ a = b
- ( ab ≠ 0 and bc ≠ 0 ) ⇒ abc ≠ 0
- If a ≠ 0 there exist unique x, y, z, such that xa = a, ay = a, za = y.
- ( e ≠ 0 and f ≠ 0 ) ⇒ eSf ≠ 0.
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C&P p. 101 |
Free semigroup FX |
- Set of finite sequences of elements of X with the operation
( x1, ..., xm ) ( y1, ..., yn ) = ( x1, ..., xm, y1, ..., yn )
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Gril p. 18 |
Rees matrix semigroup |
- G0 a group G with 0 adjoined.
- P : Λ × I → G0 a map.
- Define an operation in I × G0 × Λ by ( i, g, λ ) ( j, h, μ ) = ( i, g P( λ, j ) h, μ ).
- ( I, G0, Λ )/( I × { 0 } × Λ ) is the Rees matrix semigroup M0 ( G0; I, Λ ; P ).
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C&P p.88 |
Semigroup of linear transformations |
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C&P p.57 |
Semigroup of binary relations BX |
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C&P p.13 |
Numerical semigroup |
- 0 ∈ S ⊆ N = { 0,1,2, ... } under + .
- N - S is finite
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Delg |
Semigroup with involution (*-semigroup) |
- There exists a unary operation a → a* in S such that a** = a and (ab)* = b*a*.
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Howi |
Baer–Levi semigroup |
- Semigroup of one-to-one transformations f of X such that X − f ( X ) is infinite.
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C&P II Ch.8 |
U-semigroup |
- There exists a unary operation a → a’ in S such that ( a’)’ = a.
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Howi p.102 |
I-semigroup |
- There exists a unary operation a → a’ in S such that ( a’)’ = a and aa’a = a.
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Howi p.102 |
Semiband |
- A regular semigroup generated by its idempotents.
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Howi p.230 |
Group |
- There exists h such that for all a, ah = ha = a.
- There exists x (depending on a) such that ax = xa = h.
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Topological semigroup |
- A semigroup which is also a topological space. Such that the semigroup product is continuous.
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Pin p. 130 |
Syntactic semigroup |
- The smallest finite monoid which can recognize a subset of another semigroup.
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Pin p. 14 |
: the R-trivial monoids |
- R-trivial. That is, each R-equivalence class is trivial.
- Equivalently, for finite semigroup:
.
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Pin p. 158 |
: the L-trivial monoids |
- L-trivial. That is, each L-equivalence class is trivial.
- Equivalently, for finite monoids,
.
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Pin p. 158 |
: the J-trivial monoids |
- Monoids which are J-trivial. That is, each J-equivalence class is trivial.
- Equivalently, the monoids which are L-trivial and R-trivial.
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Pin p. 158 |
: idempotent and R-trivial monoids |
- R-trivial. That is, each R-equivalence class is trivial.
- Equivalently, for finite monoids: aba = ab.
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Pin p. 158 |
: idempotent and L-trivial monoids |
- L-trivial. That is, each L-equivalence class is trivial.
- Equivalently, for finite monoids: aba = ba.
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Pin p. 158 |
: Semigroup whose regular D are semigroup |
- Equivalently, for finite monoids:
.
- Equivalently, regular H-classes are groups,
- Equivalently, v≤Ja implies v R va and v L av
- Equivalently, for each idempotent e, the set of a such that e≤Ja is closed under product (i.e. this set is a subsemigroup)
- Equivalently, there exists no idempotent e and f such that e J f but not ef J e
- Equivalently, the monoid
does not divide 
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Pin pp. 154, 155, 158 |
: Semigroup whose regular D are aperiodic semigroup |
- Each regular D-class is an aperiodic semigroup
- Equivalently, every regular D-class is a rectangular band
- Equivalently, regular D-class are semigroup, and furthermore S is aperiodic
- Equivalently, for finite monoid: regular D-class are semigroup, and furthermore

- Equivalently, e≤Ja implies eae = e
- Equivalently, e≤Jf implies efe = e.
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Pin p. 156, 158 |
/ : Lefty trivial semigroup |
- e: eS = e,
- Equivalently, I is a left zero semigroup equal to E,
- Equivalently, for finite semigroup: I is a left zero semigroup equals
,
- Equivalently, for finite semigroup:
,
- Equivalently, for finite semigroup:
.
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Pin pp. 149, 158 |
/ : Right trivial semigroup |
- e: Se = e,
- Equivalently, I is a right zero semigroup equal to E,
- Equivalently, for finite semigroup: I is a right zero semigroup equals
,
- Equivalently, for finite semigroup:
,
- Equivalently, for finite semigroup:
.
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Pin pp. 149, 158 |
: Locally trivial semigroup |
- eSe = e,
- Equivalently, I is equal to E,
- Equivalently, eaf = ef,
- Equivalently, for finite semigroup:
,
- Equivalently, for finite semigroup:
,
- Equivalently, for finite semigroup:
.
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Pin pp. 150, 158 |
: Locally groups |
- eSe is a group,
- Equivalently, E⊆I,
- Equivalently, for finite semigroup:
.
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Pin pp. 151, 158 |