Special classes of semigroups

<h2 id="notations">Notations</h2>
In describing the defining properties of the various special classes of semigroups, the following notational conventions are adopted.

Notations<table><tbody><tr><th>Notation</th><th>Meaning</th></tr><tr><td>S</td><td>Arbitrary semigroup</td></tr><tr><td>E</td><td>Set of idempotents in S</td></tr><tr><td>G</td><td>Group of units in S</td></tr><tr><td>I</td><td>Minimal ideal of S</td></tr><tr><td>V</td><td><a href="/facts/Regular_semigroup/gQ1pytKh">Regular</a> elements of S</td></tr><tr><td>X</td><td>Arbitrary set</td></tr><tr><td>a, b, c</td><td>Arbitrary elements of S</td></tr><tr><td>x, y, z</td><td>Specific elements of S</td></tr><tr><td>e, f, g</td><td>Arbitrary elements of E</td></tr><tr><td>h</td><td>Specific element of E</td></tr><tr><td>l, m, n</td><td>Arbitrary positive integers</td></tr><tr><td>j, k</td><td>Specific positive integers</td></tr><tr><td>v, w</td><td>Arbitrary elements of V</td></tr><tr><td>0</td><td>Zero element of S</td></tr><tr><td>1</td><td>Identity element of S</td></tr><tr><td>S1</td><td>S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S</td></tr><tr><td>a ≤L ba ≤R ba ≤H ba ≤J b</td><td>S1a ⊆ S1b aS1 ⊆ bS1 S1a ⊆ S1b and aS1 ⊆ bS1 S1aS1 ⊆ S1bS1</td></tr><tr><td>L, R, H, D, J</td><td><a href="/facts/Green%27s_relations/WIcMsL7c">Green's relations</a></td></tr><tr><td>La, Ra, Ha, Da, Ja</td><td>Green classes containing a</td></tr><tr><td> x ω {\displaystyle x^{\omega }} </td><td>The only power of x which is idempotent. This element exists, assuming the semigroup is (locally) finite. See <a href="/facts/Variety_of_finite_semigroups/xn69aPHQ">variety of finite semigroups</a> for more information about this notation.</td></tr><tr><td> | X | {\displaystyle |X|} </td><td>The cardinality of X, assuming X is finite.</td></tr></tbody></table>
For example, the definition xab = xba should be read as:

<ul><li>There exists x an element of the semigroup such that, for each a and b in the semigroup, xab and xba are equal.</li></ul>
<h2 id="list-of-special-classes-of-semigroups">List of special classes of semigroups</h2>
The third column states whether this set of semigroups forms a <a href="/facts/Variety_(universal_algebra)/vXWYFyDI">variety</a>. And whether the set of finite semigroups of this special class forms a <a href="/facts/Variety_of_finite_semigroups/xn69aPHQ">variety of finite semigroups</a>. Note that if this set is a variety, its set of finite elements is automatically a variety of finite semigroups.

List of special classes of semigroups<table><tbody><tr><th>Terminology</th><th>Defining property</th><th>Variety of finite semigroup</th><th>Reference(s)</th></tr><tr><td><a href="/facts/Finite_set/za1newlP">Finite</a> semigroup</td><td><ul><li>S is a <a href="/facts/Finite_set/za1newlP">finite set</a>.</li></ul></td><td><ul><li>Not infinite</li><li>Finite</li></ul></td><td></td></tr><tr><td><a href="/facts/Empty_semigroup/UKudol1V">Empty semigroup</a></td><td><ul><li>S = ∅ {\displaystyle \emptyset } </li></ul></td><td>No</td><td></td></tr><tr><td><a href="/facts/Trivial_semigroup/33v5jsvX">Trivial semigroup</a></td><td><ul><li>Cardinality of S is 1.</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td></td></tr><tr><td><a href="/facts/Monoid/6AHve7p8">Monoid</a></td><td><ul><li>1 ∈ S</li></ul></td><td>No</td><td>Gril p. 3</td></tr><tr><td><a href="/facts/Band_(mathematics)/CdwD3R0Y">Band</a>(Idempotent semigroup)</td><td><ul><li>a2 = a</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>C&P p. 4</td></tr><tr><td><a href="/facts/Band_(mathematics)/CdwD3R0Y">Rectangular band</a></td><td><ul><li>A band such that abca = acba</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>Fennemore</td></tr><tr><td><a href="/facts/Semilattice/kW5po0rp">Semilattice</a></td><td>A commutative band, that is:<ul><li>a2 = a</li><li>ab = ba </li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td><ul><li>C&P p. 24</li><li>Fennemore</li></ul></td></tr><tr><td><a href="/facts/Commutative/WaYxJLxd">Commutative</a> semigroup</td><td><ul><li>ab = ba </li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>C&P p. 3</td></tr><tr><td><a href="/facts/Archimedean_property/61pC7UP6">Archimedean</a> commutative semigroup</td><td><ul><li>ab = ba</li><li>There exists x and k such that ak = xb.</li></ul></td><td></td><td>C&P p. 131</td></tr><tr><td><a href="/facts/Nowhere_commutative_semigroup/VmMGHtQN">Nowhere commutative semigroup</a></td><td><ul><li>ab = ba   ⇒   a = b</li></ul></td><td></td><td>C&P p. 26</td></tr><tr><td>Left weakly commutative</td><td><ul><li>There exist x and k such that (ab)k = bx.</li></ul></td><td></td><td>Nagy p. 59</td></tr><tr><td>Right weakly commutative</td><td><ul><li>There exist x and k such that (ab)k = xa.</li></ul></td><td></td><td>Nagy p. 59</td></tr><tr><td>Weakly commutative</td><td>Left and right weakly commutative. That is:<ul><li>There exist x and j such that (ab)j = bx.</li><li>There exist y and k such that (ab)k = ya.</li></ul></td><td></td><td>Nagy p. 59</td></tr><tr><td>Conditionally commutative semigroup</td><td><ul><li>If ab = ba then axb = bxa for all x.</li></ul></td><td></td><td>Nagy p. 77</td></tr><tr><td>R-commutative semigroup</td><td><ul><li>ab R ba</li></ul></td><td></td><td>Nagy p. 69–71</td></tr><tr><td>RC-commutative semigroup</td><td><ul><li>R-commutative and conditionally commutative</li></ul></td><td></td><td>Nagy p. 93–107</td></tr><tr><td>L-commutative semigroup</td><td><ul><li>ab L ba</li></ul></td><td></td><td>Nagy p. 69–71</td></tr><tr><td>LC-commutative semigroup</td><td><ul><li>L-commutative and conditionally commutative</li></ul></td><td></td><td>Nagy p. 93–107</td></tr><tr><td>H-commutative semigroup</td><td><ul><li>ab H ba</li></ul></td><td></td><td>Nagy p. 69–71</td></tr><tr><td>Quasi-commutative semigroup</td><td><ul><li>ab = (ba)k for some k.</li></ul></td><td></td><td>Nagy p. 109</td></tr><tr><td>Right commutative semigroup</td><td><ul><li>xab = xba</li></ul></td><td></td><td>Nagy p. 137</td></tr><tr><td>Left commutative semigroup</td><td><ul><li>abx = bax</li></ul></td><td></td><td>Nagy p. 137</td></tr><tr><td>Externally commutative semigroup</td><td><ul><li>axb = bxa</li></ul></td><td></td><td>Nagy p. 175</td></tr><tr><td>Medial semigroup</td><td><ul><li>xaby = xbay</li></ul></td><td></td><td>Nagy p. 119</td></tr><tr><td>E-k semigroup (k fixed)</td><td><ul><li>(ab)k = akbk</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>Nagy p. 183</td></tr><tr><td><a href="/facts/Exponential_function/ewlYxvR2">Exponential</a> semigroup</td><td><ul><li>(ab)m = ambm for all m</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>Nagy p. 183</td></tr><tr><td>WE-k semigroup (k fixed)</td><td><ul><li>There is a positive integer j depending on the couple (a,b) such that (ab)k+j = akbk (ab)j = (ab)jakbk</li></ul></td><td></td><td>Nagy p. 199</td></tr><tr><td>Weakly <a href="/facts/Exponential_function/ewlYxvR2">exponential</a> semigroup</td><td><ul><li>WE-m for all m</li></ul></td><td></td><td>Nagy p. 215</td></tr><tr><td><a href="/facts/Cancellative_semigroup/m9Q7oUxd">Right cancellative semigroup</a></td><td><ul><li>ba = ca   ⇒   b = c</li></ul></td><td></td><td>C&P p. 3</td></tr><tr><td><a href="/facts/Cancellative_semigroup/m9Q7oUxd">Left cancellative semigroup</a></td><td><ul><li>ab = ac   ⇒   b = c</li></ul></td><td></td><td>C&P p. 3</td></tr><tr><td><a href="/facts/Cancellative_semigroup/m9Q7oUxd">Cancellative semigroup</a></td><td>Left and right cancellative semigroup, that is<ul><li>ab = ac   ⇒   b = c</li><li>ba = ca   ⇒   b = c</li></ul></td><td></td><td>C&P p. 3</td></tr><tr><td>''E''-inversive semigroup (E-dense semigroup)</td><td><ul><li>There exists x such that ax ∈ E.</li></ul></td><td></td><td>C&P p. 98</td></tr><tr><td><a href="/facts/Regular_semigroup/gQ1pytKh">Regular semigroup</a></td><td><ul><li>There exists x such that axa =a.</li></ul></td><td></td><td>C&P p. 26</td></tr><tr><td><a href="/facts/Band_(mathematics)/CdwD3R0Y">Regular band</a></td><td><ul><li>A band such that abaca = abca</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>Fennemore</td></tr><tr><td>Intra-regular semigroup</td><td><ul><li>There exist x and y such that xa2y = a.</li></ul></td><td></td><td>C&P p. 121</td></tr><tr><td>Left regular semigroup</td><td><ul><li>There exists x such that xa2 = a.</li></ul></td><td></td><td>C&P p. 121</td></tr><tr><td><a href="/facts/Band_(mathematics)/CdwD3R0Y">Left-regular band</a></td><td><ul><li>A band such that aba = ab</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>Fennemore</td></tr><tr><td>Right regular semigroup</td><td><ul><li>There exists x such that a2x = a.</li></ul></td><td></td><td>C&P p. 121</td></tr><tr><td><a href="/facts/Band_(mathematics)/CdwD3R0Y">Right-regular band</a></td><td><ul><li>A band such that aba = ba</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>Fennemore</td></tr><tr><td><a href="/facts/Completely_regular_semigroup/7DGHh3Gv">Completely regular semigroup</a></td><td><ul><li>Ha is a group.</li></ul></td><td></td><td>Gril p. 75</td></tr><tr><td>(inverse) <a href="/facts/Clifford_semigroup/VycI7ub8">Clifford semigroup</a></td><td><ul><li>A regular semigroup in which all idempotents are central.</li><li>Equivalently, for finite semigroup: a ω b = b a ω {\displaystyle a^{\omega }b=ba^{\omega }} </li></ul></td><td><ul><li>Finite</li></ul></td><td>Petrich p. 65</td></tr><tr><td>k-regular semigroup (k fixed)</td><td><ul><li>There exists x such that akxak = ak.</li></ul></td><td></td><td>Hari</td></tr><tr><td>Eventually regular semigroup(π-regular semigroup, Quasi regular semigroup)</td><td><ul><li>There exists k and x (depending on a) such that akxak = ak.</li></ul></td><td></td><td>EdwaShumHigg p. 49</td></tr><tr><td>Quasi-periodic semigroup, <a href="/facts/Epigroup/IwtI3kuQ">epigroup</a>, group-bound semigroup, completely (or strongly) π-regular semigroup, and many other; see Kela for a list)</td><td><ul><li>There exists k (depending on a) such that ak belongs to a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of S</li></ul></td><td></td><td>KelaGril p. 110Higg p. 4</td></tr><tr><td>Primitive semigroup</td><td><ul><li>If 0 ≠ e and f = ef = fe then e = f.</li></ul></td><td></td><td>C&P p. 26</td></tr><tr><td>Unit regular semigroup</td><td><ul><li>There exists u in G such that aua = a.</li></ul></td><td></td><td>Tvm</td></tr><tr><td>Strongly unit regular semigroup</td><td><ul><li>There exists u in G such that aua = a.</li><li>e D f ⇒ f = v−1ev for some v in G.</li></ul></td><td></td><td>Tvm</td></tr><tr><td><a href="/facts/Orthodox_semigroup/GGrCCfHS">Orthodox semigroup</a></td><td><ul><li>There exists x such that axa = a.</li><li>E is a subsemigroup of S.</li></ul></td><td></td><td>Gril p. 57Howi p. 226</td></tr><tr><td><a href="/facts/Inverse_semigroup/gz4ezaRq">Inverse semigroup</a></td><td><ul><li>There exists unique x such that axa = a and xax = x.</li></ul></td><td></td><td>C&P p. 28</td></tr><tr><td>Left inverse semigroup (R-unipotent)</td><td><ul><li>Ra contains a unique h.</li></ul></td><td></td><td>Gril p. 382</td></tr><tr><td>Right inverse semigroup(L-unipotent)</td><td><ul><li>La contains a unique h.</li></ul></td><td></td><td>Gril p. 382</td></tr><tr><td>Locally inverse semigroup (Pseudoinverse semigroup)</td><td><ul><li>There exists x such that axa = a.</li><li>E is a pseudosemilattice.</li></ul></td><td></td><td>Gril p. 352</td></tr><tr><td>M-inversive semigroup</td><td><ul><li>There exist x and y such that baxc = bc and byac = bc.</li></ul></td><td></td><td>C&P p. 98</td></tr><tr><td>Abundant semigroup</td><td><ul><li>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.</li></ul></td><td></td><td>Chen</td></tr><tr><td>Rpp-semigroup(Right principal projective semigroup)</td><td><ul><li>The class L*a, where a L* b if ac = ad ⇔ bc = bd, contains at least one idempotent.</li></ul></td><td></td><td>Shum</td></tr><tr><td>Lpp-semigroup(Left principal projective semigroup)</td><td><ul><li>The class R*a, where a R* b if ca = da ⇔ cb = db, contains at least one idempotent.</li></ul></td><td></td><td>Shum</td></tr><tr><td><a href="/facts/Null_semigroup/eyRwHwrM">Null semigroup</a> (<a href="/facts/Zero_semigroup/eyRwHwrM">Zero semigroup</a>)</td><td><ul><li>0 ∈ S</li><li>ab = 0</li><li>Equivalently ab = cd</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>C&P p. 4</td></tr><tr><td><a href="/facts/Left_zero_semigroup/eyRwHwrM">Left zero semigroup</a></td><td><ul><li>ab = a</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>C&P p. 4</td></tr><tr><td><a href="/facts/Band_(mathematics)/CdwD3R0Y">Left zero band</a></td><td>A left zero semigroup which is a band. That is:<ul><li>ab = a</li><li>aa = a</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td><ul><li>Fennemore</li></ul></td></tr><tr><td><a href="/facts/Right_group/xG4t0wFN">Left group</a></td><td><ul><li>A semigroup which is left simple and right cancellative.</li><li>The direct product of a left zero semigroup and an abelian group.</li></ul></td><td></td><td>C&P p. 37, 38</td></tr><tr><td><a href="/facts/Right_zero_semigroup/eyRwHwrM">Right zero semigroup</a></td><td><ul><li>ab = b</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>C&P p. 4</td></tr><tr><td><a href="/facts/Band_(mathematics)/CdwD3R0Y">Right zero band</a></td><td>A right zero semigroup which is a band. That is:<ul><li>ab = b</li><li>aa = a</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>Fennemore</td></tr><tr><td><a href="/facts/Right_group/xG4t0wFN">Right group</a></td><td><ul><li>A semigroup which is right simple and left cancellative.</li><li>The direct product of a right zero semigroup and a group.</li></ul></td><td></td><td>C&P p. 37, 38</td></tr><tr><td><a href="/facts/Right_group/xG4t0wFN">Right abelian group</a></td><td><ul><li>A right simple and conditionally commutative semigroup.</li></ul><ul><li>The direct product of a right zero semigroup and an abelian group.</li></ul></td><td></td><td>Nagy p. 87</td></tr><tr><td>Unipotent semigroup</td><td><ul><li>E is singleton.</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>C&P p. 21</td></tr><tr><td>Left reductive semigroup</td><td><ul><li>If xa = xb for all x then a = b.</li></ul></td><td></td><td>C&P p. 9</td></tr><tr><td>Right reductive semigroup</td><td><ul><li>If ax = bx for all x then a = b.</li></ul></td><td></td><td>C&P p. 4</td></tr><tr><td>Reductive semigroup</td><td><ul><li>If xa = xb for all x then a = b.</li><li>If ax = bx for all x then a = b.</li></ul></td><td></td><td>C&P p. 4</td></tr><tr><td>Separative semigroup</td><td><ul><li>ab = a2 = b2   ⇒   a = b</li></ul></td><td></td><td>C&P p. 130–131</td></tr><tr><td>Reversible semigroup</td><td><ul><li>Sa ∩ Sb ≠ Ø</li><li>aS ∩ bS ≠ Ø</li></ul></td><td></td><td>C&P p. 34</td></tr><tr><td>Right reversible semigroup</td><td><ul><li>Sa ∩ Sb ≠ Ø</li></ul></td><td></td><td>C&P p. 34</td></tr><tr><td>Left reversible semigroup</td><td><ul><li>aS ∩ bS ≠ Ø</li></ul></td><td></td><td>C&P p. 34</td></tr><tr><td><a href="/facts/Aperiodic_semigroup/WaQdAsS4">Aperiodic semigroup</a></td><td><ul><li>There exists k (depending on a) such that ak = ak+1</li><li>Equivalently, for finite semigroup: for each a, a ω a = a ω {\displaystyle a^{\omega }a=a^{\omega }} .</li></ul></td><td></td><td><ul><li>KKM p. 29</li><li>Pin p. 158</li></ul></td></tr><tr><td>ω-semigroup</td><td><ul><li>E is countable descending chain under the order a ≤H b</li></ul></td><td></td><td>Gril p. 233–238</td></tr><tr><td>Left Clifford semigroup(LC-semigroup)</td><td><ul><li>aS ⊆ Sa</li></ul></td><td></td><td>Shum</td></tr><tr><td>Right Clifford semigroup(RC-semigroup)</td><td><ul><li>Sa ⊆ aS</li></ul></td><td></td><td>Shum</td></tr><tr><td>Orthogroup</td><td><ul><li>Ha is a group.</li><li>E is a subsemigroup of S</li></ul></td><td></td><td>Shum</td></tr><tr><td>Complete commutative semigroup</td><td><ul><li>ab = ba</li><li>ak is in a subgroup of S for some k.</li><li>Every nonempty subset of E has an infimum.</li></ul></td><td></td><td>Gril p. 110</td></tr><tr><td><a href="/facts/Nilsemigroup/GI8JYr7N">Nilsemigroup</a> (Nilpotent semigroup)</td><td><ul><li>0 ∈ S</li><li>ak = 0 for some integer k which depends on a.</li><li>Equivalently, for finite semigroup: for each element x and y, y x ω = x ω = x ω y {\displaystyle yx^{\omega }=x^{\omega }=x^{\omega }y} .</li></ul></td><td><ul><li>Finite</li></ul></td><td><ul><li>Gril p. 99</li><li>Pin p. 148</li></ul></td></tr><tr><td>Elementary semigroup</td><td><ul><li>ab = ba</li><li>S is of the form G ∪ N where</li><li>G is a group, and 1 ∈ G</li><li>N is an ideal, a nilsemigroup, and 0 ∈ N</li></ul></td><td></td><td>Gril p. 111</td></tr><tr><td>E-unitary semigroup</td><td><ul><li>There exists unique x such that axa = a and xax = x.</li><li>ea = e   ⇒   a ∈ E</li></ul></td><td></td><td>Gril p. 245</td></tr><tr><td>Finitely presented semigroup</td><td><ul><li>S has a <a href="/facts/Presentation_of_a_semigroup/pW4dkEq5">presentation</a> ( X; R ) in which X and R are finite.</li></ul></td><td></td><td>Gril p. 134</td></tr><tr><td>Fundamental semigroup</td><td><ul><li>Equality on S is the only congruence contained in H.</li></ul></td><td></td><td>Gril p. 88</td></tr><tr><td>Idempotent generated semigroup</td><td><ul><li>S is equal to the semigroup generated by E.</li></ul></td><td></td><td>Gril p. 328</td></tr><tr><td>Locally finite semigroup</td><td><ul><li>Every finitely generated subsemigroup of S is finite.</li></ul></td><td><ul><li>Not infinite</li><li>Finite</li></ul></td><td>Gril p. 161</td></tr><tr><td>N-semigroup</td><td><ul><li>ab = ba</li><li>There exists x and a positive integer n such that a = xbn.</li><li>ax = ay   ⇒   x = y</li><li>xa = ya   ⇒   x = y</li><li>E = Ø</li></ul></td><td></td><td>Gril p. 100</td></tr><tr><td>L-unipotent semigroup (Right inverse semigroup)</td><td><ul><li>La contains a unique e.</li></ul></td><td></td><td>Gril p. 362</td></tr><tr><td>R-unipotent semigroup (Left inverse semigroup)</td><td><ul><li>Ra contains a unique e.</li></ul></td><td></td><td>Gril p. 362</td></tr><tr><td>Left simple semigroup</td><td><ul><li>La = S</li></ul></td><td></td><td>Gril p. 57</td></tr><tr><td>Right simple semigroup</td><td><ul><li>Ra = S</li></ul></td><td></td><td>Gril p. 57</td></tr><tr><td>Subelementary semigroup</td><td><ul><li>ab = ba</li><li>S = C ∪ N where C is a cancellative semigroup, N is a nilsemigroup or a one-element semigroup.</li><li>N is ideal of S.</li><li>Zero of N is 0 of S.</li><li>For x, y in S and c in C, cx = cy implies that x = y.</li></ul></td><td></td><td>Gril p. 134</td></tr><tr><td>Symmetric semigroup(<a href="/facts/Transformation_semigroup/MGQ2QKI0">Full transformation semigroup</a>)</td><td><ul><li>Set of all mappings of X into itself with composition of mappings as binary operation.</li></ul></td><td></td><td>C&P p. 2</td></tr><tr><td>Weakly reductive semigroup</td><td><ul><li>If xz = yz and zx = zy for all z in S then x = y.</li></ul></td><td></td><td>C&P p. 11</td></tr><tr><td>Right unambiguous semigroup</td><td><ul><li>If x, y ≥R z then x ≥R y or y ≥R x.</li></ul></td><td></td><td>Gril p. 170</td></tr><tr><td>Left unambiguous semigroup</td><td><ul><li>If x, y ≥L z then x ≥L y or y ≥L x.</li></ul></td><td></td><td>Gril p. 170</td></tr><tr><td>Unambiguous semigroup</td><td><ul><li>If x, y ≥R z then x ≥R y or y ≥R x.</li><li>If x, y ≥L z then x ≥L y or y ≥L x.</li></ul></td><td></td><td>Gril p. 170</td></tr><tr><td>Left 0-unambiguous</td><td><ul><li>0∈ S</li><li>0 ≠ x ≤L y, z   ⇒   y ≤L z or z ≤L y</li></ul></td><td></td><td>Gril p. 178</td></tr><tr><td>Right 0-unambiguous</td><td><ul><li>0∈ S</li><li>0 ≠ x ≤R y, z   ⇒   y ≤L z or z ≤R y</li></ul></td><td></td><td>Gril p. 178</td></tr><tr><td>0-unambiguous semigroup</td><td><ul><li>0∈ S</li><li>0 ≠ x ≤L y, z   ⇒   y ≤L z or z ≤L y</li><li>0 ≠ x ≤R y, z   ⇒   y ≤L z or z ≤R y</li></ul></td><td></td><td>Gril p. 178</td></tr><tr><td>Left Putcha semigroup</td><td><ul><li>a ∈ bS1   ⇒   an ∈ b2S1 for some n.</li></ul></td><td></td><td>Nagy p. 35</td></tr><tr><td>Right Putcha semigroup</td><td><ul><li>a ∈ S1b   ⇒   an ∈ S1b2 for some n.</li></ul></td><td></td><td>Nagy p. 35</td></tr><tr><td>Putcha semigroup</td><td><ul><li>a ∈ S1b S1   ⇒   an ∈ S1b2S1 for some positive integer n</li></ul></td><td></td><td>Nagy p. 35</td></tr><tr><td>Bisimple semigroup(D-simple semigroup)</td><td><ul><li>Da = S</li></ul></td><td></td><td>C&P p. 49</td></tr><tr><td>0-bisimple semigroup</td><td><ul><li>0 ∈ S</li><li>S - {0} is a D-class of S.</li></ul></td><td></td><td>C&P p. 76</td></tr><tr><td>Completely simple semigroup</td><td><ul><li>There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A.</li><li>There exists h in E such that whenever hf = f and fh = f we have h = f.</li></ul></td><td></td><td>C&P p. 76</td></tr><tr><td>Completely 0-simple semigroup</td><td><ul><li>0 ∈ S</li><li>S2 ≠ 0</li><li>If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0 or A = S.</li><li>There exists non-zero h in E such that whenever hf = f, fh = f and f ≠ 0 we have h = f.</li></ul></td><td></td><td>C&P p. 76</td></tr><tr><td>D-simple semigroup(Bisimple semigroup)</td><td><ul><li>Da = S</li></ul></td><td></td><td>C&P p. 49</td></tr><tr><td>Semisimple semigroup</td><td><ul><li>Let J(a) = S1aS1, I(a) = J(a) − Ja. Each Rees factor semigroup J(a)/I(a) is 0-simple or simple.</li></ul></td><td></td><td>C&P p. 71–75</td></tr><tr><td> C S {\displaystyle \mathbf {CS} } : Simple semigroup</td><td><ul><li>Ja = S. (There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A.),</li><li>equivalently, for finite semigroup: a ω a = a {\displaystyle a^{\omega }a=a} and ( a b a ) ω = a ω {\displaystyle (aba)^{\omega }=a^{\omega }} .</li></ul></td><td><ul><li>Finite</li></ul></td><td><ul><li>C&P p. 5</li><li>Higg p. 16</li><li>Pin pp. 151, 158</li></ul></td></tr><tr><td>0-simple semigroup</td><td><ul><li>0 ∈ S</li><li>S2 ≠ 0</li><li>If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0.</li></ul></td><td></td><td>C&P p. 67</td></tr><tr><td>Left 0-simple semigroup</td><td><ul><li>0 ∈ S</li><li>S2 ≠ 0</li><li>If A ⊆ S is such that SA ⊆ A then A = 0.</li></ul></td><td></td><td>C&P p. 67</td></tr><tr><td>Right 0-simple semigroup</td><td><ul><li>0 ∈ S</li><li>S2 ≠ 0</li><li>If A ⊆ S is such that AS ⊆ A then A = 0.</li></ul></td><td></td><td>C&P p. 67</td></tr><tr><td><a href="/facts/Cyclic_semigroup/XmnoSDj6">Cyclic semigroup</a> (<a href="/facts/Monogenic_semigroup/XmnoSDj6">Monogenic semigroup</a>)</td><td><ul><li>S = { w, w2, w3, ... } for some w in S</li></ul></td><td><ul><li>Not infinite</li><li>Not finite</li></ul></td><td>C&P p. 19</td></tr><tr><td>Periodic semigroup</td><td><ul><li>{ a, a2, a3, ... } is a finite set.</li></ul></td><td><ul><li>Not infinite</li><li>Finite</li></ul></td><td>C&P p. 20</td></tr><tr><td><a href="/facts/Bicyclic_semigroup/GehG9bj5">Bicyclic semigroup</a></td><td><ul><li>1 ∈ S</li><li>S admits the <a href="/facts/Presentation_of_a_semigroup/pW4dkEq5">presentation</a> ⟨ x , y ∣ x y = 1 ⟩ {\displaystyle \langle x,y\mid xy=1\rangle } .</li></ul></td><td></td><td>C&P p. 43–46</td></tr><tr><td><a href="/facts/Full_transformation_semigroup/MGQ2QKI0">Full transformation semigroup</a> TX(Symmetric semigroup)</td><td><ul><li><a href="/facts/Set_(mathematics)/BfucfHMq">Set</a> of all <a href="/facts/Map_(mathematics)/FABF1l2f">mappings</a> of X into itself with <a href="/facts/Composition_of_functions/r5Pvnmth">composition of mappings</a> as <a href="/facts/Binary_operation/CYtow0bz">binary operation</a>.</li></ul></td><td></td><td>C&P p. 2</td></tr><tr><td><a href="/facts/Band_(mathematics)/CdwD3R0Y">Rectangular band</a></td><td><ul><li>A band such that aba = a</li><li>Equivalently abc = ac</li></ul></td><td><ul><li>Infinite</li><li>Finite</li></ul></td><td>Fennemore</td></tr><tr><td>Rectangular semigroup</td><td><ul><li>Whenever three of ax, ay, bx, by are equal, all four are equal.</li></ul></td><td></td><td>C&P p. 97</td></tr><tr><td><a href="/facts/Symmetric_inverse_semigroup/c8NXPzqM">Symmetric inverse semigroup</a> IX</td><td><ul><li>The semigroup of <a href="/facts/Bijection/j4gTuTmW">one-to-one</a> <a href="/facts/Partial_function/0vJMw0Nm">partial transformations</a> of X.</li></ul></td><td></td><td>C&P p. 29</td></tr><tr><td><a href="/facts/Brandt_semigroup/NcDbeq8H">Brandt semigroup</a></td><td><ul><li>0 ∈ S</li><li>( ac = bc ≠ 0 or ca = cb ≠ 0 )   ⇒   a = b</li><li>( ab ≠ 0 and bc ≠ 0 )   ⇒   abc ≠ 0</li><li>If a ≠ 0 there exist unique x, y, z, such that xa = a, ay = a, za = y.</li><li>( e ≠ 0 and f ≠ 0 )   ⇒   eSf ≠ 0.</li></ul></td><td></td><td>C&P p. 101</td></tr><tr><td><a href="/facts/Free_semigroup/h3nPVjqr">Free semigroup</a> FX</td><td><ul><li>Set of finite sequences of elements of X with the operation( x1, ..., xm ) ( y1, ..., yn ) = ( x1, ..., xm, y1, ..., yn )</li></ul></td><td></td><td>Gril p. 18</td></tr><tr><td>Rees <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> semigroup</td><td><ul><li>G0 a group G with 0 adjoined.</li><li>P : Λ × I → G0 a map.</li><li>Define an operation in I × G0 × Λ by ( i, g, λ ) ( j, h, μ ) = ( i, g P( λ, j ) h, μ ).</li><li>( I, G0, Λ )/( I × { 0 } × Λ ) is the Rees matrix semigroup M0 ( G0; I, Λ ; P ).</li></ul></td><td></td><td>C&P p.88</td></tr><tr><td>Semigroup of <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a></td><td><ul><li>Semigroup of <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a> of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> F under <a href="/facts/Composition_of_functions/r5Pvnmth">composition of functions</a>.</li></ul></td><td></td><td>C&P p.57</td></tr><tr><td>Semigroup of <a href="/facts/Binary_relation/r9BwGgaK">binary relations</a> BX</td><td><ul><li>Set of all <a href="/facts/Binary_relation/r9BwGgaK">binary relations</a> on X under <a href="/facts/Composition_of_relations/tD8joBlo">composition</a></li></ul></td><td></td><td>C&P p.13</td></tr><tr><td><a href="/facts/Numerical_semigroup/5LrBPgmX">Numerical semigroup</a></td><td><ul><li>0 ∈ S ⊆ N = { 0,1,2, ... } under + .</li><li>N - S is finite</li></ul></td><td></td><td>Delg</td></tr><tr><td><a href="/facts/Semigroup_with_involution/aXJEUAmY">Semigroup with involution</a>(*-semigroup)</td><td><ul><li>There exists a unary operation a → a* in S such that a** = a and (ab)* = b*a*.</li></ul></td><td></td><td>Howi</td></tr><tr><td>Baer–Levi semigroup</td><td><ul><li>Semigroup of one-to-one transformations f of X such that X − f ( X ) is infinite.</li></ul></td><td></td><td>C&P II Ch.8</td></tr><tr><td><a href="/facts/U-semigroup/CwMygHSj">U-semigroup</a></td><td><ul><li>There exists a unary operation a → a’ in S such that ( a’)’ = a.</li></ul></td><td></td><td>Howi p.102</td></tr><tr><td><a href="/facts/I-semigroup/CwMygHSj">I-semigroup</a></td><td><ul><li>There exists a unary operation a → a’ in S such that ( a’)’ = a and aa’a = a.</li></ul></td><td></td><td>Howi p.102</td></tr><tr><td>Semiband</td><td><ul><li>A regular semigroup generated by its idempotents.</li></ul></td><td></td><td>Howi p.230</td></tr><tr><td><a href="/facts/Group_(mathematics)/IQTYqINt">Group</a></td><td><ul><li>There exists h such that for all a, ah = ha = a.</li><li>There exists x (depending on a) such that ax = xa = h.</li></ul></td><td><ul><li>Not infinite</li><li>Finite</li></ul></td><td></td></tr><tr><td><a href="/facts/Topological_semigroup/JDwmyd5M">Topological semigroup</a></td><td><ul><li>A semigroup which is also a topological space. Such that the semigroup product is continuous.</li></ul></td><td><ul><li>Not applicable</li></ul></td><td>Pin p. 130</td></tr><tr><td><a href="/facts/Syntactic_semigroup/7khpKalF">Syntactic semigroup</a></td><td><ul><li>The smallest finite monoid which can <a href="/facts/Recognizable_set/UP5IRBDQ">recognize</a> a subset of another semigroup.</li></ul></td><td></td><td>Pin p. 14</td></tr><tr><td> R {\displaystyle \mathbf {R} } : the R-trivial monoids</td><td><ul><li>R-trivial. That is, each R-equivalence class is trivial.</li><li>Equivalently, for finite semigroup: ( a b ) ω a = ( a b ) ω {\displaystyle (ab)^{\omega }a=(ab)^{\omega }} .</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin p. 158</td></tr><tr><td> L {\displaystyle \mathbf {L} } : the L-trivial monoids</td><td><ul><li>L-trivial. That is, each L-equivalence class is trivial.</li><li>Equivalently, for finite monoids, b ( a b ) ω = ( a b ) ω {\displaystyle b(ab)^{\omega }=(ab)^{\omega }} .</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin p. 158</td></tr><tr><td> J {\displaystyle \mathbf {J} } : the J-trivial monoids</td><td><ul><li>Monoids which are J-trivial. That is, each J-equivalence class is trivial.</li><li>Equivalently, the monoids which are L-trivial and R-trivial.</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin p. 158</td></tr><tr><td> R 1 {\displaystyle \mathbf {R_{1}} } : idempotent and R-trivial monoids</td><td><ul><li>R-trivial. That is, each R-equivalence class is trivial.</li><li>Equivalently, for finite monoids: aba = ab.</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin p. 158</td></tr><tr><td> L 1 {\displaystyle \mathbf {L_{1}} } : idempotent and L-trivial monoids</td><td><ul><li>L-trivial. That is, each L-equivalence class is trivial.</li><li>Equivalently, for finite monoids: aba = ba.</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin p. 158</td></tr><tr><td> D S {\displaystyle \mathbb {D} \mathbf {S} } : Semigroup whose regular D are semigroup</td><td><ul><li>Equivalently, for finite monoids: ( a ω a ω a ω ) ω = a ω {\displaystyle (a^{\omega }a^{\omega }a^{\omega })^{\omega }=a^{\omega }} .</li><li>Equivalently, regular H-classes are groups,</li><li>Equivalently, v≤Ja implies v R va and v L av</li><li>Equivalently, for each idempotent e, the set of a such that e≤Ja is closed under product (i.e. this set is a subsemigroup)</li><li>Equivalently, there exists no idempotent e and f such that e J f but not ef J e</li><li>Equivalently, the monoid B 2 1 {\displaystyle B_{2}^{1}} does not divide S × S {\displaystyle S\times S} </li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 154, 155, 158</td></tr><tr><td> D A {\displaystyle \mathbb {D} \mathbf {A} } : Semigroup whose regular D are aperiodic semigroup</td><td><ul><li>Each regular D-class is an aperiodic semigroup</li><li>Equivalently, every regular D-class is a rectangular band</li><li>Equivalently, regular D-class are semigroup, and furthermore S is aperiodic</li><li>Equivalently, for finite monoid: regular D-class are semigroup, and furthermore a a ω = a ω {\displaystyle aa^{\omega }=a^{\omega }} </li><li>Equivalently, e≤Ja implies eae = e</li><li>Equivalently, e≤Jf implies efe = e.</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin p. 156, 158</td></tr><tr><td> ℓ 1 {\displaystyle \ell \mathbf {1} } / K {\displaystyle \mathbf {K} } : Lefty trivial semigroup</td><td><ul><li>e: eS = e,</li><li>Equivalently, I is a left zero semigroup equal to E,</li><li>Equivalently, for finite semigroup: I is a left zero semigroup equals S | S | {\displaystyle S^{|S|}} ,</li><li>Equivalently, for finite semigroup: a 1 … a n y = a 1 … a n {\displaystyle a_{1}\dots a_{n}y=a_{1}\dots a_{n}} ,</li><li>Equivalently, for finite semigroup: a ω b = a ω {\displaystyle a^{\omega }b=a^{\omega }} .</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 149, 158</td></tr><tr><td> r 1 {\displaystyle \mathbf {r1} } / D {\displaystyle \mathbf {D} } : Right trivial semigroup</td><td><ul><li>e: Se = e,</li><li>Equivalently, I is a right zero semigroup equal to E,</li><li>Equivalently, for finite semigroup: I is a right zero semigroup equals S | S | {\displaystyle S^{|S|}} ,</li><li>Equivalently, for finite semigroup: b a 1 … a n = a 1 … a n {\displaystyle ba_{1}\dots a_{n}=a_{1}\dots a_{n}} ,</li><li>Equivalently, for finite semigroup: b a ω = a ω {\displaystyle ba^{\omega }=a^{\omega }} .</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 149, 158</td></tr><tr><td> L 1 {\displaystyle \mathbb {L} \mathbf {1} } : Locally trivial semigroup</td><td><ul><li>eSe = e,</li><li>Equivalently, I is equal to E,</li><li>Equivalently, eaf = ef,</li><li>Equivalently, for finite semigroup: y a 1 … a n = a 1 … a n {\displaystyle ya_{1}\dots a_{n}=a_{1}\dots a_{n}} ,</li><li>Equivalently, for finite semigroup: a 1 … a n y a 1 … a n = a 1 … a n {\displaystyle a_{1}\dots a_{n}ya_{1}\dots a_{n}=a_{1}\dots a_{n}} ,</li><li>Equivalently, for finite semigroup: a ω b a ω = a ω {\displaystyle a^{\omega }ba^{\omega }=a^{\omega }} .</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 150, 158</td></tr><tr><td> L G {\displaystyle \mathbb {L} \mathbf {G} } : Locally groups</td><td><ul><li>eSe is a group,</li><li>Equivalently, E⊆I,</li><li>Equivalently, for finite semigroup: ( a ω b a ω ) ω = a ω {\displaystyle (a^{\omega }ba^{\omega })^{\omega }=a^{\omega }} .</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 151, 158</td></tr></tbody></table>
List of special classes of ordered semigroups<table><tbody><tr><td></td></tr></tbody><tbody><tr><th>Terminology</th><th>Defining property</th><th>Variety</th><th>Reference(s)</th></tr><tr><td><a href="/facts/Ordered_semigroup/AeNaNxPo">Ordered semigroup</a></td><td><ul><li>A semigroup with a partial order relation ≤, such that a ≤ b implies c•a ≤ c•b and a•c ≤ b•c</li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin p. 14</td></tr><tr><td> N + {\displaystyle \mathbf {N} ^{+}} </td><td><ul><li>Nilpotent finite semigroups, with a ≤ b ω {\displaystyle a\leq b^{\omega }} </li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 157, 158</td></tr><tr><td> N − {\displaystyle \mathbf {N} ^{-}} </td><td><ul><li>Nilpotent finite semigroups, with b ω ≤ a {\displaystyle b^{\omega }\leq a} </li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 157, 158</td></tr><tr><td> J 1 + {\displaystyle \mathbf {J} _{1}^{+}} </td><td><ul><li>Semilattices with 1 ≤ a {\displaystyle 1\leq a} </li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 157, 158</td></tr><tr><td> J 1 − {\displaystyle \mathbf {J} _{1}^{-}} </td><td><ul><li>Semilattices with a ≤ 1 {\displaystyle a\leq 1} </li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 157, 158</td></tr><tr><td> L J 1 + {\displaystyle \mathbb {L} \mathbf {J} _{1}^{+}} locally positive J-trivial semigroup</td><td><ul><li>Finite semigroups satisfying a ω ≤ a ω b a ω {\displaystyle a^{\omega }\leq a^{\omega }ba^{\omega }} </li></ul></td><td><ul><li>Finite</li></ul></td><td>Pin pp. 157, 158</td></tr></tbody></table>

<table><tbody><tr><td>[C&P]</td><td></td><td><a href="/facts/A._H._Clifford/6jpc6VPA">A. H. Clifford</a>, <a href="/facts/G._B._Preston/xWLdPXcE">G. B. Preston</a> (1964). The Algebraic Theory of Semigroups Vol. I (Second Edition). <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-0272-4</td></tr><tr><td>[C&P II]  </td><td></td><td>A. H. Clifford, G. B. Preston (1967). The Algebraic Theory of Semigroups Vol. II (Second Edition). <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0272-0</td></tr><tr><td>[Chen] </td><td></td><td>Hui Chen (2006), "Construction of a kind of abundant semigroups", Mathematical Communications (11), 165–171 (Accessed on 25 April 2009)</td></tr><tr><td>[Delg]</td><td></td><td>M. Delgado, et al., Numerical semigroups, <a href="http://www.gap-system.org/Manuals/pkg/numericalsgps/doc/manual.pdf">[1]</a> (Accessed on 27 April 2009)</td></tr><tr><td>[Edwa]</td><td></td><td>P. M. Edwards (1983), "Eventually regular semigroups", Bulletin of Australian Mathematical Society 28, 23–38</td></tr><tr><td>[Gril]</td><td></td><td>P. A. Grillet (1995). Semigroups. <a href="/facts/CRC_Press/duTuH4hz">CRC Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8247-9662-4</td></tr><tr><td>[Hari]</td><td></td><td>K. S. Harinath (1979), "Some results on k-regular semigroups", Indian Journal of Pure and Applied Mathematics 10(11), 1422–1431</td></tr><tr><td>[Howi]</td><td></td><td><a href="/facts/John_Mackintosh_Howie/M9eL8Fsy">J. M. Howie</a> (1995), Fundamentals of Semigroup Theory, <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a></td></tr><tr><td>[Nagy]</td><td></td><td>Attila Nagy (2001). Special Classes of Semigroups. <a href="/facts/Springer_Science%2BBusiness_Media/nAesf6nT">Springer</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7923-6890-8</td></tr><tr><td>[Pet]</td><td></td><td> M. Petrich, N. R. Reilly (1999). Completely regular semigroups. <a href="/facts/John_Wiley_%26_Sons/53g3LqJc">John Wiley & Sons</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-19571-9</td></tr><tr><td>[Shum]    </td><td></td><td>K. P. Shum "Rpp semigroups, its generalizations and special subclasses" in Advances in Algebra and Combinatorics edited by K P Shum et al. (2008), <a href="/facts/World_Scientific/qJPTP0Yk">World Scientific</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 981-279-000-4 (pp. 303–334)</td></tr><tr><td>[Tvm]</td><td></td><td>Proceedings of the International Symposium on Theory of Regular Semigroups and Applications, <a href="/facts/University_of_Kerala/5k6lp2Qb">University of Kerala</a>, <a href="/facts/Thiruvananthapuram/WzEGT3qx">Thiruvananthapuram</a>, <a href="/facts/India/1kKHcnxM">India</a>, 1986</td></tr><tr><td>[Kela]</td><td></td><td>A. V. Kelarev, Applications of epigroups to graded ring theory, <a href="/facts/Semigroup_Forum/T5pQJlKS">Semigroup Forum</a>, Volume 50, Number 1 (1995), 327-350 <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02573530">10.1007/BF02573530</a></td></tr><tr><td>[KKM]</td><td></td><td>Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev (2000), Monoids, Acts and Categories: with Applications to Wreath Products and Graphs, Expositions in Mathematics 29, Walter de Gruyter, Berlin, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-11-015248-7.</td></tr><tr><td>[Higg]</td><td></td><td> Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-853577-5.</td></tr><tr><td>[Pin]</td><td></td><td><a href="/facts/Jean-%C3%89ric_Pin/hBjkW0FN">Pin, Jean-Éric</a> (2016-11-30). <a href="http://www.liafa.jussieu.fr/~jep/PDF/MPRI/MPRI.pdf">Mathematical Foundations of Automata Theory</a> (PDF).</td></tr><tr><td>[Fennemore]</td><td></td><td>Fennemore, Charles (1970), "All varieties of bands", <a href="/facts/Semigroup_Forum/T5pQJlKS">Semigroup Forum</a>, 1 (1): 172–179, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02573031">10.1007/BF02573031</a></td></tr></tbody></table>

Special classes of semigroups open-in-new

Special classes of semigroups