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In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ab = ba for all elements a and b in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively.

In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of subsets of the underlying set. The underlying sets are not assumed to carry any other mathematical structures like order or topology.

As in any algebraic theory, one of the main problems of the theory of semigroups is the classification of all semigroups and a complete description of their structure. In the case of semigroups, since the binary operation is required to satisfy only the associativity property the problem of classification is considered extremely difficult. Descriptions of structures have been obtained for certain special classes of semigroups. For example, the structure of the sets of idempotents of regular semigroups is completely known. Structure descriptions are presented in terms of better known types of semigroups. The best known type of semigroup is the group.

A (necessarily incomplete) list of various special classes of semigroups is presented below. To the extent possible the defining properties are formulated in terms of the binary operations in the semigroups. The references point to the locations from where the defining properties are sourced.

• Notations

• List of special classes of semigroups

• References

In describing the defining properties of the various special classes of semigroups, the following notational conventions are adopted.

| Notation | Meaning |

|---|---|

| S | Arbitrary semigroup |

| E | Set of idempotents in S |

| G | Group of units in S |

| I | Minimal ideal of S |

| V | Regular elements of S |

| X | Arbitrary set |

| a , b , c | Arbitrary elements of S |

| x , y , z | Specific elements of S |

| e , f , g | Arbitrary elements of E |

| h | Specific element of E |

| l , m , n | Arbitrary positive integers |

| j , k | Specific positive integers |

| v , w | Arbitrary elements of V |

| 0 | Zero element of S |

| 1 | Identity element of S |

| S 1 | S if 1 ∈ S ; S ∪ { 1 } if 1 ∉ S |

| a ≤ L b a ≤ R b a ≤ H b a ≤ J b | S 1 a ⊆ S 1 b aS 1 ⊆ bS 1 S 1 a ⊆ S 1 b… |

| L , R , H , D , J | Green's relations |

| L a , R a , H a , D a , J a | Green classes containing a |

| x ω {\displaystyle x^{\omega }} | The only power of x which is idempotent… |

| / X / {\displaystyle /X/} | The cardinality of X , assuming X is fi… |

For example, the definition xab = xba should be read as:

• There exists x an element of the semigroup such that, for each a and b in the semigroup, xab and xba are equal.

The third column states whether this set of semigroups forms a variety. And whether the set of finite semigroups of this special class forms a variety of finite semigroups. Note that if this set is a variety, its set of finite elements is automatically a variety of finite semigroups.

| Terminology | Defining property |

|---|---|

| Finite semigroup | S is a finite set . |

| Empty semigroup | S = ∅ {\displaystyle \emptyset } |

| Trivial semigroup | Cardinality of S is 1. |

| Monoid | 1 ∈ S |

| Band (Idempotent semigroup) | a 2 = a |

| Rectangular band | A band such that abca = acba |

| Semilattice | A commutative band, that is: a 2 = a ab… |

| Commutative semigroup | ab = ba |

| Archimedean commutative semigroup | ab = ba There exists x and k such that… |

| Nowhere commutative semigroup | ab = ba ⇒ a = b |

| Left weakly commutative | There exist x and k such that ( ab ) k… |

| Right weakly commutative | There exist x and k such that ( ab ) k… |

| Weakly commutative | Left and right weakly commutative. That… |

| Conditionally commutative semigroup | If ab = ba then axb = bxa for all x . |

| R -commutative semigroup | ab R ba |

| RC -commutative semigroup | R -commutative and conditionally commut… |

| L -commutative semigroup | ab L ba |

| LC -commutative semigroup | L -commutative and conditionally commut… |

| H -commutative semigroup | ab H ba |

| Quasi-commutative semigroup | ab = ( ba ) k for some k . |

| Right commutative semigroup | xab = xba |

| Left commutative semigroup | abx = bax |

| Externally commutative semigroup | axb = bxa |

| Medial semigroup | xaby = xbay |

| E- k semigroup ( k fixed) | ( ab ) k = a k b k |

| Exponential semigroup | ( ab ) m = a m b m for all m |

| WE- k semigroup ( k fixed) | There is a positive integer j depending… |

| Weakly exponential semigroup | WE- m for all m |

| Right cancellative semigroup | ba = ca ⇒ b = c |

| Left cancellative semigroup | ab = ac ⇒ b = c |

| Cancellative semigroup | Left and right cancellative semigroup,… |

| E -inversive semigroup ( E -dense semig… | There exists x such that ax ∈ E . |

| Regular semigroup | There exists x such that axa = a . |

| Regular band | A band such that abaca = abca |

| Intra-regular semigroup | There exist x and y such that xa 2 y =… |

| Left regular semigroup | There exists x such that xa 2 = a . |

| Left-regular band | A band such that aba = ab |

| Right regular semigroup | There exists x such that a 2 x = a . |

| Right-regular band | A band such that aba = ba |

| Completely regular semigroup | H a is a group. |

| (inverse) Clifford semigroup | A regular semigroup in which all idempo… |

| k -regular semigroup ( k fixed) | There exists x such that a k xa k = a k… |

| Eventually regular semigroup (π-regular… | There exists k and x (depending on a )… |

| Quasi-periodic semigroup, epigroup , gr… | There exists k (depending on a ) such t… |

| Primitive semigroup | If 0 ≠ e and f = ef = fe then e = f . |

| Unit regular semigroup | There exists u in G such that aua = a . |

| Strongly unit regular semigroup | There exists u in G such that aua = a .… |

| Orthodox semigroup | There exists x such that axa = a . E is… |

| Inverse semigroup | There exists unique x such that axa = a… |

| Left inverse semigroup ( R -unipotent) | R a contains a unique h . |

| Right inverse semigroup ( L -unipotent) | L a contains a unique h . |

| Locally inverse semigroup (Pseudoinvers… | There exists x such that axa = a . E is… |

| M -inversive semigroup | There exist x and y such that baxc = bc… |

| Abundant semigroup | The classes L * a and R * a , where a L… |

| Rpp-semigroup (Right principal projecti… | The class L * a , where a L * b if ac =… |

| Lpp-semigroup (Left principal projectiv… | The class R * a , where a R * b if ca =… |

| Null semigroup ( Zero semigroup ) | 0 ∈ S ab = 0 Equivalently ab = cd |

| Left zero semigroup | ab = a |

| Left zero band | A left zero semigroup which is a band.… |

| Left group | A semigroup which is left simple and ri… |

| Right zero semigroup | ab = b |

| Right zero band | A right zero semigroup which is a band.… |

| Right group | A semigroup which is right simple and l… |

| Right abelian group | A right simple and conditionally commut… |

| Unipotent semigroup | E is singleton. |

| Left reductive semigroup | If xa = xb for all x then a = b . |

| Right reductive semigroup | If ax = bx for all x then a = b . |

| Reductive semigroup | If xa = xb for all x then a = b . If ax… |

| Separative semigroup | ab = a 2 = b 2 ⇒ a = b |

| Reversible semigroup | Sa ∩ Sb ≠ Ø aS ∩ bS ≠ Ø |

| Right reversible semigroup | Sa ∩ Sb ≠ Ø |

| Left reversible semigroup | aS ∩ bS ≠ Ø |

| Aperiodic semigroup | There exists k (depending on a ) such t… |

| ω-semigroup | E is countable descending chain under t… |

| Left Clifford semigroup (LC-semigroup) | aS ⊆ Sa |

| Right Clifford semigroup (RC-semigroup) | Sa ⊆ aS |

| Orthogroup | H a is a group. E is a subsemigroup of S |

| Complete commutative semigroup | ab = ba a k is in a subgroup of S for s… |

| Nilsemigroup (Nilpotent semigroup) | 0 ∈ S a k = 0 for some integer k which… |

| Elementary semigroup | ab = ba S is of the form G ∪ N where G… |

| E -unitary semigroup | There exists unique x such that axa = a… |

| Finitely presented semigroup | S has a presentation ( X ; R ) in which… |

| Fundamental semigroup | Equality on S is the only congruence co… |

| Idempotent generated semigroup | S is equal to the semigroup generated b… |

| Locally finite semigroup | Every finitely generated subsemigroup o… |

| N -semigroup | ab = ba There exists x and a positive i… |

| L -unipotent semigroup (Right inverse s… | L a contains a unique e . |

| R -unipotent semigroup (Left inverse se… | R a contains a unique e . |

| Left simple semigroup | L a = S |

| Right simple semigroup | R a = S |

| Subelementary semigroup | ab = ba S = C ∪ N where C is a cancella… |

| Symmetric semigroup ( Full transformati… | Set of all mappings of X into itself wi… |

| Weakly reductive semigroup | If xz = yz and zx = zy for all z in S t… |

| Right unambiguous semigroup | If x , y ≥ R z then x ≥ R y or y ≥ R x . |

| Left unambiguous semigroup | If x , y ≥ L z then x ≥ L y or y ≥ L x . |

| Unambiguous semigroup | If x , y ≥ R z then x ≥ R y or y ≥ R x… |

| Left 0-unambiguous | 0∈ S 0 ≠ x ≤ L y , z ⇒ y ≤ L z or z ≤ L… |

| Right 0-unambiguous | 0∈ S 0 ≠ x ≤ R y , z ⇒ y ≤ L z or z ≤ R… |

| 0-unambiguous semigroup | 0∈ S 0 ≠ x ≤ L y , z ⇒ y ≤ L z or z ≤ L… |

| Left Putcha semigroup | a ∈ bS 1 ⇒ a n ∈ b 2 S 1 for some n . |

| Right Putcha semigroup | a ∈ S 1 b ⇒ a n ∈ S 1 b 2 for some n . |

| Putcha semigroup | a ∈ S 1 b S 1 ⇒ a n ∈ S 1 b 2 S 1 for s… |

| Bisimple semigroup ( D -simple semigrou… | D a = S |

| 0-bisimple semigroup | 0 ∈ S S - {0} is a D -class of S . |

| Completely simple semigroup | There exists no A ⊆ S , A ≠ S such that… |

| Completely 0-simple semigroup | 0 ∈ S S 2 ≠ 0 If A ⊆ S is such that AS… |

| D -simple semigroup (Bisimple semigroup) | D a = S |

| Semisimple semigroup | Let J ( a ) = S 1 aS 1 , I ( a ) = J (… |

| C S {\displaystyle \mathbf {CS} } : S… | J a = S . (There exists no A ⊆ S , A ≠… |

| 0-simple semigroup | 0 ∈ S S 2 ≠ 0 If A ⊆ S is such that AS… |

| Left 0-simple semigroup | 0 ∈ S S 2 ≠ 0 If A ⊆ S is such that SA… |

| Right 0-simple semigroup | 0 ∈ S S 2 ≠ 0 If A ⊆ S is such that AS… |

| Cyclic semigroup ( Monogenic semigroup ) | S = { w , w 2 , w 3 , ... } for some w… |

| Periodic semigroup | { a , a 2 , a 3 , ... } is a finite set. |

| Bicyclic semigroup | 1 ∈ S S admits the presentation ⟨ x , y… |

| Full transformation semigroup T X (Symm… | Set of all mappings of X into itself wi… |

| Rectangular band | A band such that aba = a Equivalently a… |

| Rectangular semigroup | Whenever three of ax , ay , bx , by are… |

| Symmetric inverse semigroup I X | The semigroup of one-to-one partial tra… |

| Brandt semigroup | 0 ∈ S ( ac = bc ≠ 0 or ca = cb ≠ 0 ) ⇒… |

| Free semigroup F X | Set of finite sequences of elements of… |

| Rees matrix semigroup | G 0 a group G with 0 adjoined. P : Λ ×… |

| Semigroup of linear transformations | Semigroup of linear transformations of… |

| Semigroup of binary relations B X | Set of all binary relations on X under… |

| Numerical semigroup | 0 ∈ S ⊆ N = { 0,1,2, ... } under + . N… |

| Semigroup with involution (*-semigroup) | There exists a unary operation a → a *… |

| Baer–Levi semigroup | Semigroup of one-to-one transformations… |

| U -semigroup | There exists a unary operation a → a ’… |

| I -semigroup | There exists a unary operation a → a ’… |

| Semiband | A regular semigroup generated by its id… |

| Group | There exists h such that for all a, ah… |

| Topological semigroup | A semigroup which is also a topological… |

| Syntactic semigroup | The smallest finite monoid which can re… |

| R {\displaystyle \mathbf {R} } : the… | R -trivial. That is, each R -equivalenc… |

| L {\displaystyle \mathbf {L} } : the… | L -trivial. That is, each L -equivalenc… |

| J {\displaystyle \mathbf {J} } : the… | Monoids which are J -trivial. That is,… |

| R 1 {\displaystyle \mathbf {R_{1}} }… | R -trivial. That is, each R -equivalenc… |

| L 1 {\displaystyle \mathbf {L_{1}} }… | L -trivial. That is, each L -equivalenc… |

| D S {\displaystyle \mathbb {D} \math… | Equivalently, for finite monoids: ( a ω… |

| D A {\displaystyle \mathbb {D} \math… | Each regular D-class is an aperiodic se… |

| ℓ 1 {\displaystyle \ell \mathbf {1}… | e : eS = e , Equivalently, I is a left… |

| r 1 {\displaystyle \mathbf {r1} } / D… | e : Se = e , Equivalently, I is a right… |

| L 1 {\displaystyle \mathbb {L} \math… | eSe = e , Equivalently, I is equal to E… |

| L G {\displaystyle \mathbb {L} \math… | eSe is a group, Equivalently, E ⊆ I , E… |

| Terminology | Variety of finite semigroup |

|---|---|

| Finite semigroup | Not infinite Finite |

| Empty semigroup | No |

| Trivial semigroup | Infinite Finite |

| Monoid | No |

| Band (Idempotent semigroup) | Infinite Finite |

| Rectangular band | Infinite Finite |

| Semilattice | Infinite Finite |

| Commutative semigroup | Infinite Finite |

| E- k semigroup ( k fixed) | Infinite Finite |

| Exponential semigroup | Infinite Finite |

| Regular band | Infinite Finite |

| Left-regular band | Infinite Finite |

| Right-regular band | Infinite Finite |

| (inverse) Clifford semigroup | Finite |

| Null semigroup ( Zero semigroup ) | Infinite Finite |

| Left zero semigroup | Infinite Finite |

| Left zero band | Infinite Finite |

| Right zero semigroup | Infinite Finite |

| Right zero band | Infinite Finite |

| Unipotent semigroup | Infinite Finite |

| Nilsemigroup (Nilpotent semigroup) | Finite |

| Locally finite semigroup | Not infinite Finite |

| C S {\displaystyle \mathbf {CS} } : S… | Finite |

| Cyclic semigroup ( Monogenic semigroup ) | Not infinite Not finite |

| Periodic semigroup | Not infinite Finite |

| Rectangular band | Infinite Finite |

| Group | Not infinite Finite |

| Topological semigroup | Not applicable |

| R {\displaystyle \mathbf {R} } : the… | Finite |

| L {\displaystyle \mathbf {L} } : the… | Finite |

| J {\displaystyle \mathbf {J} } : the… | Finite |

| R 1 {\displaystyle \mathbf {R_{1}} }… | Finite |

| L 1 {\displaystyle \mathbf {L_{1}} }… | Finite |

| D S {\displaystyle \mathbb {D} \math… | Finite |

| D A {\displaystyle \mathbb {D} \math… | Finite |

| ℓ 1 {\displaystyle \ell \mathbf {1}… | Finite |

| r 1 {\displaystyle \mathbf {r1} } / D… | Finite |

| L 1 {\displaystyle \mathbb {L} \math… | Finite |

| L G {\displaystyle \mathbb {L} \math… | Finite |

| Terminology | Reference(s) |

|---|---|

| Monoid | Gril p. 3 |

| Band (Idempotent semigroup) | C&P p. 4 |

| Rectangular band | Fennemore |

| Semilattice | C&P p. 24 Fennemore |

| Commutative semigroup | C&P p. 3 |

| Archimedean commutative semigroup | C&P p. 131 |

| Nowhere commutative semigroup | C&P p. 26 |

| Left weakly commutative | Nagy p. 59 |

| Right weakly commutative | Nagy p. 59 |

| Weakly commutative | Nagy p. 59 |

| Conditionally commutative semigroup | Nagy p. 77 |

| R -commutative semigroup | Nagy p. 69–71 |

| RC -commutative semigroup | Nagy p. 93–107 |

| L -commutative semigroup | Nagy p. 69–71 |

| LC -commutative semigroup | Nagy p. 93–107 |

| H -commutative semigroup | Nagy p. 69–71 |

| Quasi-commutative semigroup | Nagy p. 109 |

| Right commutative semigroup | Nagy p. 137 |

| Left commutative semigroup | Nagy p. 137 |

| Externally commutative semigroup | Nagy p. 175 |

| Medial semigroup | Nagy p. 119 |

| E- k semigroup ( k fixed) | Nagy p. 183 |

| Exponential semigroup | Nagy p. 183 |

| WE- k semigroup ( k fixed) | Nagy p. 199 |

| Weakly exponential semigroup | Nagy p. 215 |

| Right cancellative semigroup | C&P p. 3 |

| Left cancellative semigroup | C&P p. 3 |

| Cancellative semigroup | C&P p. 3 |

| E -inversive semigroup ( E -dense semig… | C&P p. 98 |

| Regular semigroup | C&P p. 26 |

| Regular band | Fennemore |

| Intra-regular semigroup | C&P p. 121 |

| Left regular semigroup | C&P p. 121 |

| Left-regular band | Fennemore |

| Right regular semigroup | C&P p. 121 |

| Right-regular band | Fennemore |

| Completely regular semigroup | Gril p. 75 |

| (inverse) Clifford semigroup | Petrich p. 65 |

| k -regular semigroup ( k fixed) | Hari |

| Eventually regular semigroup (π-regular… | Edwa Shum Higg p. 49 |

| Quasi-periodic semigroup, epigroup , gr… | Kela Gril p. 110 Higg p. 4 |

| Primitive semigroup | C&P p. 26 |

| Unit regular semigroup | Tvm |

| Strongly unit regular semigroup | Tvm |

| Orthodox semigroup | Gril p. 57 Howi p. 226 |

| Inverse semigroup | C&P p. 28 |

| Left inverse semigroup ( R -unipotent) | Gril p. 382 |

| Right inverse semigroup ( L -unipotent) | Gril p. 382 |

| Locally inverse semigroup (Pseudoinvers… | Gril p. 352 |

| M -inversive semigroup | C&P p. 98 |

| Abundant semigroup | Chen |

| Rpp-semigroup (Right principal projecti… | Shum |

| Lpp-semigroup (Left principal projectiv… | Shum |

| Null semigroup ( Zero semigroup ) | C&P p. 4 |

| Left zero semigroup | C&P p. 4 |

| Left zero band | Fennemore |

| Left group | C&P p. 37, 38 |

| Right zero semigroup | C&P p. 4 |

| Right zero band | Fennemore |

| Right group | C&P p. 37, 38 |

| Right abelian group | Nagy p. 87 |

| Unipotent semigroup | C&P p. 21 |

| Left reductive semigroup | C&P p. 9 |

| Right reductive semigroup | C&P p. 4 |

| Reductive semigroup | C&P p. 4 |

| Separative semigroup | C&P p. 130–131 |

| Reversible semigroup | C&P p. 34 |

| Right reversible semigroup | C&P p. 34 |

| Left reversible semigroup | C&P p. 34 |

| Aperiodic semigroup | KKM p. 29 Pin p. 158 |

| ω-semigroup | Gril p. 233–238 |

| Left Clifford semigroup (LC-semigroup) | Shum |

| Right Clifford semigroup (RC-semigroup) | Shum |

| Orthogroup | Shum |

| Complete commutative semigroup | Gril p. 110 |

| Nilsemigroup (Nilpotent semigroup) | Gril p. 99 Pin p. 148 |

| Elementary semigroup | Gril p. 111 |

| E -unitary semigroup | Gril p. 245 |

| Finitely presented semigroup | Gril p. 134 |

| Fundamental semigroup | Gril p. 88 |

| Idempotent generated semigroup | Gril p. 328 |

| Locally finite semigroup | Gril p. 161 |

| N -semigroup | Gril p. 100 |

| L -unipotent semigroup (Right inverse s… | Gril p. 362 |

| R -unipotent semigroup (Left inverse se… | Gril p. 362 |

| Left simple semigroup | Gril p. 57 |

| Right simple semigroup | Gril p. 57 |

| Subelementary semigroup | Gril p. 134 |

| Symmetric semigroup ( Full transformati… | C&P p. 2 |

| Weakly reductive semigroup | C&P p. 11 |

| Right unambiguous semigroup | Gril p. 170 |

| Left unambiguous semigroup | Gril p. 170 |

| Unambiguous semigroup | Gril p. 170 |

| Left 0-unambiguous | Gril p. 178 |

| Right 0-unambiguous | Gril p. 178 |

| 0-unambiguous semigroup | Gril p. 178 |

| Left Putcha semigroup | Nagy p. 35 |

| Right Putcha semigroup | Nagy p. 35 |

| Putcha semigroup | Nagy p. 35 |

| Bisimple semigroup ( D -simple semigrou… | C&P p. 49 |

| 0-bisimple semigroup | C&P p. 76 |

| Completely simple semigroup | C&P p. 76 |

| Completely 0-simple semigroup | C&P p. 76 |

| D -simple semigroup (Bisimple semigroup) | C&P p. 49 |

| Semisimple semigroup | C&P p. 71–75 |

| C S {\displaystyle \mathbf {CS} } : S… | C&P p. 5 Higg p. 16 Pin pp. 151, 158 |

| 0-simple semigroup | C&P p. 67 |

| Left 0-simple semigroup | C&P p. 67 |

| Right 0-simple semigroup | C&P p. 67 |

| Cyclic semigroup ( Monogenic semigroup ) | C&P p. 19 |

| Periodic semigroup | C&P p. 20 |

| Bicyclic semigroup | C&P p. 43–46 |

| Full transformation semigroup T X (Symm… | C&P p. 2 |

| Rectangular band | Fennemore |

| Rectangular semigroup | C&P p. 97 |

| Symmetric inverse semigroup I X | C&P p. 29 |

| Brandt semigroup | C&P p. 101 |

| Free semigroup F X | Gril p. 18 |

| Rees matrix semigroup | C&P p.88 |

| Semigroup of linear transformations | C&P p.57 |

| Semigroup of binary relations B X | C&P p.13 |

| Numerical semigroup | Delg |

| Semigroup with involution (*-semigroup) | Howi |

| Baer–Levi semigroup | C&P II Ch.8 |

| U -semigroup | Howi p.102 |

| I -semigroup | Howi p.102 |

| Semiband | Howi p.230 |

| Topological semigroup | Pin p. 130 |

| Syntactic semigroup | Pin p. 14 |

| R {\displaystyle \mathbf {R} } : the… | Pin p. 158 |

| L {\displaystyle \mathbf {L} } : the… | Pin p. 158 |

| J {\displaystyle \mathbf {J} } : the… | Pin p. 158 |

| R 1 {\displaystyle \mathbf {R_{1}} }… | Pin p. 158 |

| L 1 {\displaystyle \mathbf {L_{1}} }… | Pin p. 158 |

| D S {\displaystyle \mathbb {D} \math… | Pin pp. 154, 155, 158 |

| D A {\displaystyle \mathbb {D} \math… | Pin p. 156, 158 |

| ℓ 1 {\displaystyle \ell \mathbf {1}… | Pin pp. 149, 158 |

| r 1 {\displaystyle \mathbf {r1} } / D… | Pin pp. 149, 158 |

| L 1 {\displaystyle \mathbb {L} \math… | Pin pp. 150, 158 |

| L G {\displaystyle \mathbb {L} \math… | Pin pp. 151, 158 |

| Terminology | Defining property | Variety |

|---|---|---|

| Ordered semigroup | A semigroup with a partial order relati… | Finite |

| N + {\displaystyle \mathbf {N} ^{+}} | Nilpotent finite semigroups, with a ≤ b… | Finite |

| N − {\displaystyle \mathbf {N} ^{-}} | Nilpotent finite semigroups, with b ω ≤… | Finite |

| J 1 + {\displaystyle \mathbf {J} _{1}… | Semilattices with 1 ≤ a {\displaystyle… | Finite |

| J 1 − {\displaystyle \mathbf {J} _{1}… | Semilattices with a ≤ 1 {\displaystyle… | Finite |

| L J 1 + {\displaystyle \mathbb {L} \… | Finite semigroups satisfying a ω ≤ a ω… | Finite |

| Terminology | Reference(s) |

|---|---|

| Ordered semigroup | Pin p. 14 |

| N + {\displaystyle \mathbf {N} ^{+}} | Pin pp. 157, 158 |

| N − {\displaystyle \mathbf {N} ^{-}} | Pin pp. 157, 158 |

| J 1 + {\displaystyle \mathbf {J} _{1}… | Pin pp. 157, 158 |

| J 1 − {\displaystyle \mathbf {J} _{1}… | Pin pp. 157, 158 |

| L J 1 + {\displaystyle \mathbb {L} \… | Pin pp. 157, 158 |