The q-theory of Finite Semigroups
Citations
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1,322 citations
109 citations
Cites background or methods or result from "The q-theory of Finite Semigroups"
...This last section will be more demanding of the reader in terms of semigroup theoretic background, but most of the necessary background can be found in [4,11,18, 38 ]....
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...This is an example of one of Green’s relations [11, 15, 18, 38 ]....
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...The book of Almeida [3] contains more modern results, as does the forthcoming book [ 38 ]....
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...If s, t are J -equivalent regular elements, then there are (regular) elements x, y, u, v (J -equivalent to s and t) such that xsy = t and utv = s [11, 18, 38 ]....
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...Alternatively, one can easily verify that each generalized group mapping image of S corresponding to a regular J-class acts by partial permutations on its 0-minimal ideal [18, 38 ]....
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References
9,254 citations
"The q-theory of Finite Semigroups" refers background in this paper
...This is one of the central notions of category theory; see [177]....
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...For the present text, it is handy to be familiar with a little bit of category theory [177], basic topology [152], and perhaps some combinatorial group theory [176]....
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...We remark that in [177] the left adjoint is drawn on top, while we place it on the bottom....
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...For these reasons we establish in detail the categorical and Galois connection terminology of [177] and also the theory of continuous lattices, as expounded in [92] (see also [142, Chpt....
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...That is, when the right adjoint to ∆ exists, and there is a terminal object, we say that C has finite products; see [177]....
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5,518 citations
"The q-theory of Finite Semigroups" refers background in this paper
...The “monomial map” [108,136,369], going back to the work of Frobenius on induced representations, then placesG inside the iterated wreath product of these simple group divisors....
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3,454 citations
"The q-theory of Finite Semigroups" refers background in this paper
...For the present text, it is handy to be familiar with a little bit of category theory [177], basic topology [152], and perhaps some combinatorial group theory [176]....
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...If we orient Γ and denote by E the set of positively oriented edges, then π1(Γ, v) is freely generated by the homotopy classes of the paths peιepeτ , where e ∈ E \T [176,307,311]....
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...It is well known that π1(Γ, v) is a free group [176, 311]....
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...The reader is referred to [176,311] for more details....
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...Since the free group on A is residually a p-group [176, 209], there is an A-generated group G ∈ Gp such that any two words in A of length n−1 are sent to distinct elements of G....
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3,340 citations
"The q-theory of Finite Semigroups" refers background in this paper
...Recall that a compact Hausdorff space is totally disconnected if and only if it has a basis of simultaneously open and closed, that is clopen, subsets for its topology [112,142,152]....
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...In fact it is well known that every compact metric space is a continuous quotient of a Cantor set [152], so the above example is just the tip of the iceberg....
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...For the present text, it is handy to be familiar with a little bit of category theory [177], basic topology [152], and perhaps some combinatorial group theory [176]....
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2,750 citations
"The q-theory of Finite Semigroups" refers background or methods in this paper
...459 7.1 Birkhoff’s Subdirect Representation Theorem . . . . . . . . . . . . . . . 460 7.1.1 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 7.1.2 Elementary examples of smi decompositions . . . . . . . . . . 468 7.2 Locally Dually Algebraic Lattices . . . . . . . . . . . . . . . . . . . . . . . . . ....
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...Imagine what techniques will be needed to address the general case! We then entered into the so-called “abstract spectral theory” of lattices [92], a sweeping generalization of Stone’s duality between Boolean algebras and profinite spaces [59, 92, 112, 142, 330, 331]....
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...This is the classical Stone duality for Boolean algebras [59,112,142,330,331]....
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...Recall that a Boolean algebra [59, 92, 112, 142] is a distributive lattice L with top 1 and bottom 0 such that each element a has a complement ¬a satisfying a ∧ ¬a = 0 and a ∨ ¬a = 1 One can view L as a semiring by taking ∨ as addition and ∧ as multiplication (semirings are the subject of our next chapter)....
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...It is a well-known theorem of Birkhoff [59] that...
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