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Papers by John Fountain
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1981
SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups... more SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroup S we study the minimum good congruence γ such that S/γ is adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1981
SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups... more SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroupSwe study the minimum good congruence γ such thatS/γis adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.
This is the second in a series of papers that develops the theory of reflection monoids, motivate... more This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in arXiv:0812.2789. In this paper we study their presentations as abstract monoids. Along the way we also find general presentations for certain join-semilattices (as monoids under join) which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spin-off is a general presentation for the Renner monoid of an algebraic monoid, which we illustrate in the special case of the "classical" algebraic monoids.
International Journal of Algebra and Computation, 2009
We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse mo... more We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y*T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X* acts doubly on the semilattice [Formula: see text] of idempotents of FIM(X) and that FAM(X) is embedded in [Formula: see text]. Finally we show that every weakly E-ample monoid has a proper ample cover.
Periodica Mathematica Hungarica, 2009
Portugaliae Mathematica, 1994
Commun Algebra, 1999
The relation R on a monoid S provides a natural generalisation of Green's relation R. If every R-... more The relation R on a monoid S provides a natural generalisation of Green's relation R. If every R-class of S contains an idempotent, S is left semiabundant; if R is a left congruence then S satisfies (CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy (CL). However, the class of left semiabundant monoids is much larger, as we illustrate with a number of examples. This is the first of three related papers exploring the relationship between unipotent monoids and left semiabundancy. We consider the situations where the power enlargement or the Szendrei expansion of a monoid yields a left semiabundant monoid with (CL). Using the Szendrei expansion and the notion of the least unipotent monoid congruence σ on a 1
ABSTRACT A monoid M is an extension of a submonoid T by a group G if there is a morphism from M o... more ABSTRACT A monoid M is an extension of a submonoid T by a group G if there is a morphism from M onto G such that T is the inverse image of the identity of G. Our first main theorem gives descriptions of such extensions in terms of groups acting on categories.The theory developed is also used to obtain a second main theorem which answers the following question. Given a monoid M and a submonoid T, under what conditions can we find a monoid and a morphism θ from onto M such that is an extension of a submonoid by a group and θ maps isomorphically onto T.These results can be viewed as generalisations of two seminal theorems of McAlister in inverse semigroup theory. They are also closely related to Ash's celebrated solution of the Rhodes conjecture in finite semigroup theory.McAlister proved that each inverse monoid admits an E-unitary inverse cover and gave a structure theorem for E-unitary inverse monoids. Many researchers have extended one or both of these results to wider classes of semigroups. Almost all these generalisations can be recovered from our two main theorems.
This is the first of a series of papers in which we initiate and develop the theory of reflection... more This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.
Advances in Algebra and Combinatorics - Proceedings of the Second International Congress in Algebra and Combinatorics, 2008
Let L S denote the language of (right) S-acts over a monoid S and let Σ S be a set of sentences i... more Let L S denote the language of (right) S-acts over a monoid S and let Σ S be a set of sentences in L S which axiomatises S-acts. A general result of model theory says that Σ S has a model companion, denoted by T S , precisely when the class E of existentially closed S-acts is axiomatisable and in this case, T S axiomatises E. It is known that T S exists if and only if S is right coherent. Moreover, by a result of Ivanov, T S has the model-theoretic property of being stable. In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over T S algebraically, thus reducing our examination of T S to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that T S is stable and to prove another result of Ivanov, namely that T S is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that T S is totally transcendental and is such that the U-rank of any type coincides with its Morley rank.
Fountain, Gould and Smith introduced the concept of equivalence of orders in a semigroup and the ... more Fountain, Gould and Smith introduced the concept of equivalence of orders in a semigroup and the notion of a maximal order. We examine these ideas in the context of orders in completely 0-simple semigroups with particular emphasis on abundant orders.
Semigroups, Algorithms, Automata and Languages, 2002
Lecture Notes in Mathematics, 1988
ABSTRACT
We continue our development of the theory of reflection monoi ds by first deriving a presentation... more We continue our development of the theory of reflection monoi ds by first deriving a presentation for a general reflection monoid from a result of Easdown, East and Fitzgerald for factorizable inverse monoids. We then derive "Popova" style presentations for reflection mon oids built from Boolean hyperplane arrangements and reflection arrangements.
Semigroups and Languages - Proceedings of the Workshop, 2004
Suppose that the semigroup S is isomorphic to a subemigroup of I<SUB>X via an isomorphism. ... more Suppose that the semigroup S is isomorphic to a subemigroup of I<SUB>X via an isomorphism. We say that S is inverse if S is closed under-,, that S is left ample if S is closed under, and that S is right ample if S is closed under,. An ample semigroup is one which is both left and right ample. Note that any inverse semigroup is ample. Left ample semigroups used to be known as left type A semigroups. As usual, E(S) denotes the set of all idempotents of a semigroup S. It is immediate from the definition that, in a left ample semigroup S, the idempotents commute with each other, and so E(S) is a subsemilattice of S. A left ample semigroup satisfies the left ample condition, and similarly, a right ample semigroup the left ample condition. The two ample conditions are the properties that underly much of the structure theory for inverse semigroups, and so it is reasonable to expect that analogous structure results hold for left ample semigroups.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1981
SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups... more SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroup S we study the minimum good congruence γ such that S/γ is adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1981
SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups... more SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroupSwe study the minimum good congruence γ such thatS/γis adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.
This is the second in a series of papers that develops the theory of reflection monoids, motivate... more This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in arXiv:0812.2789. In this paper we study their presentations as abstract monoids. Along the way we also find general presentations for certain join-semilattices (as monoids under join) which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spin-off is a general presentation for the Renner monoid of an algebraic monoid, which we illustrate in the special case of the "classical" algebraic monoids.
International Journal of Algebra and Computation, 2009
We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse mo... more We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y*T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X* acts doubly on the semilattice [Formula: see text] of idempotents of FIM(X) and that FAM(X) is embedded in [Formula: see text]. Finally we show that every weakly E-ample monoid has a proper ample cover.
Periodica Mathematica Hungarica, 2009
Portugaliae Mathematica, 1994
Commun Algebra, 1999
The relation R on a monoid S provides a natural generalisation of Green's relation R. If every R-... more The relation R on a monoid S provides a natural generalisation of Green's relation R. If every R-class of S contains an idempotent, S is left semiabundant; if R is a left congruence then S satisfies (CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy (CL). However, the class of left semiabundant monoids is much larger, as we illustrate with a number of examples. This is the first of three related papers exploring the relationship between unipotent monoids and left semiabundancy. We consider the situations where the power enlargement or the Szendrei expansion of a monoid yields a left semiabundant monoid with (CL). Using the Szendrei expansion and the notion of the least unipotent monoid congruence σ on a 1
ABSTRACT A monoid M is an extension of a submonoid T by a group G if there is a morphism from M o... more ABSTRACT A monoid M is an extension of a submonoid T by a group G if there is a morphism from M onto G such that T is the inverse image of the identity of G. Our first main theorem gives descriptions of such extensions in terms of groups acting on categories.The theory developed is also used to obtain a second main theorem which answers the following question. Given a monoid M and a submonoid T, under what conditions can we find a monoid and a morphism θ from onto M such that is an extension of a submonoid by a group and θ maps isomorphically onto T.These results can be viewed as generalisations of two seminal theorems of McAlister in inverse semigroup theory. They are also closely related to Ash's celebrated solution of the Rhodes conjecture in finite semigroup theory.McAlister proved that each inverse monoid admits an E-unitary inverse cover and gave a structure theorem for E-unitary inverse monoids. Many researchers have extended one or both of these results to wider classes of semigroups. Almost all these generalisations can be recovered from our two main theorems.
This is the first of a series of papers in which we initiate and develop the theory of reflection... more This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.
Advances in Algebra and Combinatorics - Proceedings of the Second International Congress in Algebra and Combinatorics, 2008
Let L S denote the language of (right) S-acts over a monoid S and let Σ S be a set of sentences i... more Let L S denote the language of (right) S-acts over a monoid S and let Σ S be a set of sentences in L S which axiomatises S-acts. A general result of model theory says that Σ S has a model companion, denoted by T S , precisely when the class E of existentially closed S-acts is axiomatisable and in this case, T S axiomatises E. It is known that T S exists if and only if S is right coherent. Moreover, by a result of Ivanov, T S has the model-theoretic property of being stable. In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over T S algebraically, thus reducing our examination of T S to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that T S is stable and to prove another result of Ivanov, namely that T S is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that T S is totally transcendental and is such that the U-rank of any type coincides with its Morley rank.
Fountain, Gould and Smith introduced the concept of equivalence of orders in a semigroup and the ... more Fountain, Gould and Smith introduced the concept of equivalence of orders in a semigroup and the notion of a maximal order. We examine these ideas in the context of orders in completely 0-simple semigroups with particular emphasis on abundant orders.
Semigroups, Algorithms, Automata and Languages, 2002
Lecture Notes in Mathematics, 1988
ABSTRACT
We continue our development of the theory of reflection monoi ds by first deriving a presentation... more We continue our development of the theory of reflection monoi ds by first deriving a presentation for a general reflection monoid from a result of Easdown, East and Fitzgerald for factorizable inverse monoids. We then derive "Popova" style presentations for reflection mon oids built from Boolean hyperplane arrangements and reflection arrangements.
Semigroups and Languages - Proceedings of the Workshop, 2004
Suppose that the semigroup S is isomorphic to a subemigroup of I<SUB>X via an isomorphism. ... more Suppose that the semigroup S is isomorphic to a subemigroup of I<SUB>X via an isomorphism. We say that S is inverse if S is closed under-,, that S is left ample if S is closed under, and that S is right ample if S is closed under,. An ample semigroup is one which is both left and right ample. Note that any inverse semigroup is ample. Left ample semigroups used to be known as left type A semigroups. As usual, E(S) denotes the set of all idempotents of a semigroup S. It is immediate from the definition that, in a left ample semigroup S, the idempotents commute with each other, and so E(S) is a subsemilattice of S. A left ample semigroup satisfies the left ample condition, and similarly, a right ample semigroup the left ample condition. The two ample conditions are the properties that underly much of the structure theory for inverse semigroups, and so it is reasonable to expect that analogous structure results hold for left ample semigroups.