Sydney Bulman-fleming - Academia.edu (original) (raw)
Papers by Sydney Bulman-fleming
Semigroup Forum, Feb 14, 2008
If S is a monoid, a right S-act A S is a set A; equipped with a "right S-action" A S ! A sending ... more If S is a monoid, a right S-act A S is a set A; equipped with a "right S-action" A S ! A sending the pair (a; s) 2 A S to as; that sat-is…es the conditions (i) a(st) = (as)t and (ii) a1 = a for all a 2 A and s; t 2 S: If, in addition, S is equipped with a compatible partial order and A is a poset, such that the action is monotone (when A S is equipped with the product order), then A S is called a right S-poset. Left S-acts and S-posets are de…ned analogously. For a given S-act (resp. S-poset) a tensor product functor A S from left S-acts to sets (resp. left S-posets to posets) exists, and A S is called pullback ‡at or equalizer ‡at (resp. subpullback ‡at or subequalizer ‡at) if this functor preserves pullbacks or equalizers (resp. subpullbacks or subequalizers). By analogy with the Lazard-Govorov Theorem for Rmodules, B. Stenström proved in 1971 that an S-act is isomorphic to a directed colimit of …nitely generated free S-acts if and only if it is both pullback ‡at and equalizer ‡at. Some 20 years later, the present author showed that, in fact, pullback ‡atness by itself is su¢ cient. (A new, more direct proof of that result is contained in the present article.) In 2005, Valdis Laan and the present author obtained a version of the Lazard-Govorov Theorem for S-posets, in which subpullbacks
Communications in Algebra, Jan 26, 2005
For a monoid S, a (left) S-act is a nonempty set B together with a mapping S×B→B sending (s, b) t... more For a monoid S, a (left) S-act is a nonempty set B together with a mapping S×B→B sending (s, b) to sb such that S(tb) = lpar;st)b and 1b = b for all S, t ∈ S and B ∈ B. Right S-acts A can also be defined, and a tensor product A ⊗ sB (a set)can be defined that has the customary universal property with respect to balanced maps
Algebra Universalis, Dec 1, 1974
Without Abstract
Semigroup Forum, Dec 1, 1986
ABSTRACT Without Abstract
Algebra Universalis, Jun 1, 2004
Abstract.Ježek and Kepka [4] proved that a universal algebra A with at least one at least binary ... more Abstract.Ježek and Kepka [4] proved that a universal algebra A with at least one at least binary operation is isomorphic to the factor of a subdirectly irreducible algebra B by its monolith if and only if the intersection of all of its (nonempty) ideals is nonempty, and that B may be chosen to be finite if A is finite. (By an ideal of A is meant a non-empty subset I of A such that f(a1, . . . , an) ∈ I whenever f is an n-ary fundamental operation of A and a1, . . . , an ∈ A are elements with ai ∈ I for at least one index i.) In the present paper, we prove that if A is a semigroup, then B may be chosen also to be a semigroup, but that a finite semigroup need not be isomorphic to the factor of a finite subdirectly irreducible semigroup by its monolith.
American Mathematical Monthly, Aug 1, 1990
Communications in Algebra, 1991
Algebra Universalis, Dec 1, 1974
The answer to Problem 1 is 'no'. One may take 3r = ag| W', with q/the variety of groups and ~ a v... more The answer to Problem 1 is 'no'. One may take 3r = ag| W', with q/the variety of groups and ~ a variety with no finite models which does not have modular congruences, e.g. the bi-unary variety defined by the laws fgfg2x=x andfgf2g2x=fgf2g2y (S. Burris).
American Mathematical Monthly, Dec 1, 1990
Proceedings of the American Mathematical Society, Nov 1, 1978
Let S (respectively So) denote the category of all join-semilattices (resp. join-semilattices wit... more Let S (respectively So) denote the category of all join-semilattices (resp. join-semilattices with 0) with (0-preserving) semilattice homomorphisms. For A G S let A0 represent the object of S0 obtained by adjoining a new 0-element. In either category the tensor product of two objects may be constructed in such a manner that the tensor product functor is left adjoint to the hom functor. An object A eS (Sq) is called flat if the functor-®gA (-OE^) preserves monomorphisms in S (So). Theorem. For A 6 S (S0) the following conditions are equivalent: (1) A is fiat in S (Srj), (2) A0 (A) is distributive {see Grätzer, Lattice theory,p. 117), (3) A is a directed colimit of a system of f.g. free algebras in S (So). The equivalence of (1) and (2) in S was previously known to James A. Anderson. (1) «=> (3) is an analogue of Lazard's well-known result for Ä-modules.
Semigroup Forum, Dec 1, 1985
ABSTRACT Without Abstract
Mathematische Zeitschrift, 1971
Let 92=(A; F ) be a universal algebra (see, for example, [3]) and let ~-be a set of congruence re... more Let 92=(A; F ) be a universal algebra (see, for example, [3]) and let ~-be a set of congruence relations on 92 which in addition is the base of a filter [2] on A x A. The topology (resp. uniform structure) on A which 5 naturally gives rise to will be called a congruence topology (resp. congruence structure), and it is a simple matter to see that under this topology each fundamental operation of 92 is continuous. 5 will be called a congruence basis on 92, and 92 itself will be known as a congruence topological algebra, or more specifically, a J-algebra. In case (~ J = co, where co denotes the relation of equality on A, the word Hausdorff will be appended to the above-introduced terms. In the following discussion the main purpose is to construct for a given congruence topological algebra a Hausdorff completion which possesses a certain universal property. While such a construction is included among the results of H~imisch [4], an essentially shorter and more direct development seems justified, the present one being a natural generalization of a well-known construction (see [6, 7, 5], and [1]) for topological groups. For interest we also include a brief recollection of a method, due to H~imisch, of endowing arbitrary universal algebras with congruence structures. For suggesting this study and for invaluable discussions pertaining thereto the author expresses deep gratitude to Wenzel.
Semigroup Forum, 1999
... basic concepts in semigroup theory, in particular completely (0) simple semigroups and their... more ... basic concepts in semigroup theory, in particular completely (0) simple semigroups and theirRees matrix representations ... In the commutative diagram ,/ id a <2> f E ^Л , A$ ® sm -XAS ® SN, id a ... Bentz and Bulman-Fleming from the flatness of A we know that the mapping idA ...
Proceedings of the American Mathematical Society, Dec 1, 1987
Proceedings of the American Mathematical Society, Aug 1, 1985
A semigroup S is called (left, right) absolutely flat if all of its (left, right) 5-sets are flat... more A semigroup S is called (left, right) absolutely flat if all of its (left, right) 5-sets are flat. S is a (left, right) generalized inverse semigroup if S is régulai and its set of idempotents E(S) is a (left, right) normal band (i.e. a strong semilattice of (left zero, right zero) rectangular bands). In this paper it is proved that a generalized inverse semigroup S is left absolutely flat if and only if S is a right generalized inverse semigroup and the (nonidentity) structure maps of E(S) are constant. In particular all inverse semigroups are left (and right) absolutely flat (see [1]). Other consequences are derived.
Algebra Universalis, Dec 1, 1972
Mathematische Nachrichten, Dec 1, 2005
Page 1. Math. Nachr. 278, No. 15, 1743 1755 (2005) / DOI 10.1002/mana.200310338 Lazard's T... more Page 1. Math. Nachr. 278, No. 15, 1743 1755 (2005) / DOI 10.1002/mana.200310338 Lazard's Theorem for S-posets Sydney BulmanFleming ∗1 and Valdis Laan2 1 Wilfrid Laurier University, Waterloo, Canada N2L 3C5 2 University of Tartu, Tartu, Estonia ...
Semigroup Forum, Feb 1, 2004
In Comm. Algebra 30 (3) (2002), 1475-1498, Bulman-Fleming and Kilp developed various notions of f... more In Comm. Algebra 30 (3) (2002), 1475-1498, Bulman-Fleming and Kilp developed various notions of flatness of a right act A S over a monoid S that are based on the extent to which the functor A S ⊗ − preserves equalizers. In Semigroup Forum 65 (3) (2002), 428-449, Bulman-Fleming discussed in detail one of these notions, annihilator-flatness. The present paper is devoted to two more of these notions, weak equalizer-flatness and strong torsion-freeness. An act A S is called weakly equalizer-flat if the functor A S ⊗ − 'almost' preserves equalizers of any two homomorphisms into the left act S S, and strongly torsionfree if this functor 'almost' preserves equalizers of any two homomorphisms from S S into the Rees factor act S (S/Sc), where c is any right cancellable element of S. (The adverb 'almost' signifies that the canonical morphism provided by the universal property of equalizers may be only a monomorphism rather than an isomorphism.) From the definitions it is clear that flatness implies weak equalizer-flatness, which in turn implies annihilator-flatness, and it was known already that both of these implications are strict. A monoid is called right absolutely weakly equalizer-flat if all of its right acts are weakly equalizerflat. In this paper we prove a result which implies that right PP monoids with central idempotents are absolutely weakly equalizer-flat. We also show that for a relatively large class of commutative monoids, right absolute equalizer-flatness and right absolute annihilator-flatness coincide. Finally, we provide examples showing that the implication between strong torsion-freeness and torsion-freeness is strict.
Semigroup Forum, Feb 14, 2008
If S is a monoid, a right S-act A S is a set A; equipped with a "right S-action" A S ! A sending ... more If S is a monoid, a right S-act A S is a set A; equipped with a "right S-action" A S ! A sending the pair (a; s) 2 A S to as; that sat-is…es the conditions (i) a(st) = (as)t and (ii) a1 = a for all a 2 A and s; t 2 S: If, in addition, S is equipped with a compatible partial order and A is a poset, such that the action is monotone (when A S is equipped with the product order), then A S is called a right S-poset. Left S-acts and S-posets are de…ned analogously. For a given S-act (resp. S-poset) a tensor product functor A S from left S-acts to sets (resp. left S-posets to posets) exists, and A S is called pullback ‡at or equalizer ‡at (resp. subpullback ‡at or subequalizer ‡at) if this functor preserves pullbacks or equalizers (resp. subpullbacks or subequalizers). By analogy with the Lazard-Govorov Theorem for Rmodules, B. Stenström proved in 1971 that an S-act is isomorphic to a directed colimit of …nitely generated free S-acts if and only if it is both pullback ‡at and equalizer ‡at. Some 20 years later, the present author showed that, in fact, pullback ‡atness by itself is su¢ cient. (A new, more direct proof of that result is contained in the present article.) In 2005, Valdis Laan and the present author obtained a version of the Lazard-Govorov Theorem for S-posets, in which subpullbacks
Communications in Algebra, Jan 26, 2005
For a monoid S, a (left) S-act is a nonempty set B together with a mapping S×B→B sending (s, b) t... more For a monoid S, a (left) S-act is a nonempty set B together with a mapping S×B→B sending (s, b) to sb such that S(tb) = lpar;st)b and 1b = b for all S, t ∈ S and B ∈ B. Right S-acts A can also be defined, and a tensor product A ⊗ sB (a set)can be defined that has the customary universal property with respect to balanced maps
Algebra Universalis, Dec 1, 1974
Without Abstract
Semigroup Forum, Dec 1, 1986
ABSTRACT Without Abstract
Algebra Universalis, Jun 1, 2004
Abstract.Ježek and Kepka [4] proved that a universal algebra A with at least one at least binary ... more Abstract.Ježek and Kepka [4] proved that a universal algebra A with at least one at least binary operation is isomorphic to the factor of a subdirectly irreducible algebra B by its monolith if and only if the intersection of all of its (nonempty) ideals is nonempty, and that B may be chosen to be finite if A is finite. (By an ideal of A is meant a non-empty subset I of A such that f(a1, . . . , an) ∈ I whenever f is an n-ary fundamental operation of A and a1, . . . , an ∈ A are elements with ai ∈ I for at least one index i.) In the present paper, we prove that if A is a semigroup, then B may be chosen also to be a semigroup, but that a finite semigroup need not be isomorphic to the factor of a finite subdirectly irreducible semigroup by its monolith.
American Mathematical Monthly, Aug 1, 1990
Communications in Algebra, 1991
Algebra Universalis, Dec 1, 1974
The answer to Problem 1 is 'no'. One may take 3r = ag| W', with q/the variety of groups and ~ a v... more The answer to Problem 1 is 'no'. One may take 3r = ag| W', with q/the variety of groups and ~ a variety with no finite models which does not have modular congruences, e.g. the bi-unary variety defined by the laws fgfg2x=x andfgf2g2x=fgf2g2y (S. Burris).
American Mathematical Monthly, Dec 1, 1990
Proceedings of the American Mathematical Society, Nov 1, 1978
Let S (respectively So) denote the category of all join-semilattices (resp. join-semilattices wit... more Let S (respectively So) denote the category of all join-semilattices (resp. join-semilattices with 0) with (0-preserving) semilattice homomorphisms. For A G S let A0 represent the object of S0 obtained by adjoining a new 0-element. In either category the tensor product of two objects may be constructed in such a manner that the tensor product functor is left adjoint to the hom functor. An object A eS (Sq) is called flat if the functor-®gA (-OE^) preserves monomorphisms in S (So). Theorem. For A 6 S (S0) the following conditions are equivalent: (1) A is fiat in S (Srj), (2) A0 (A) is distributive {see Grätzer, Lattice theory,p. 117), (3) A is a directed colimit of a system of f.g. free algebras in S (So). The equivalence of (1) and (2) in S was previously known to James A. Anderson. (1) «=> (3) is an analogue of Lazard's well-known result for Ä-modules.
Semigroup Forum, Dec 1, 1985
ABSTRACT Without Abstract
Mathematische Zeitschrift, 1971
Let 92=(A; F ) be a universal algebra (see, for example, [3]) and let ~-be a set of congruence re... more Let 92=(A; F ) be a universal algebra (see, for example, [3]) and let ~-be a set of congruence relations on 92 which in addition is the base of a filter [2] on A x A. The topology (resp. uniform structure) on A which 5 naturally gives rise to will be called a congruence topology (resp. congruence structure), and it is a simple matter to see that under this topology each fundamental operation of 92 is continuous. 5 will be called a congruence basis on 92, and 92 itself will be known as a congruence topological algebra, or more specifically, a J-algebra. In case (~ J = co, where co denotes the relation of equality on A, the word Hausdorff will be appended to the above-introduced terms. In the following discussion the main purpose is to construct for a given congruence topological algebra a Hausdorff completion which possesses a certain universal property. While such a construction is included among the results of H~imisch [4], an essentially shorter and more direct development seems justified, the present one being a natural generalization of a well-known construction (see [6, 7, 5], and [1]) for topological groups. For interest we also include a brief recollection of a method, due to H~imisch, of endowing arbitrary universal algebras with congruence structures. For suggesting this study and for invaluable discussions pertaining thereto the author expresses deep gratitude to Wenzel.
Semigroup Forum, 1999
... basic concepts in semigroup theory, in particular completely (0) simple semigroups and their... more ... basic concepts in semigroup theory, in particular completely (0) simple semigroups and theirRees matrix representations ... In the commutative diagram ,/ id a <2> f E ^Л , A$ ® sm -XAS ® SN, id a ... Bentz and Bulman-Fleming from the flatness of A we know that the mapping idA ...
Proceedings of the American Mathematical Society, Dec 1, 1987
Proceedings of the American Mathematical Society, Aug 1, 1985
A semigroup S is called (left, right) absolutely flat if all of its (left, right) 5-sets are flat... more A semigroup S is called (left, right) absolutely flat if all of its (left, right) 5-sets are flat. S is a (left, right) generalized inverse semigroup if S is régulai and its set of idempotents E(S) is a (left, right) normal band (i.e. a strong semilattice of (left zero, right zero) rectangular bands). In this paper it is proved that a generalized inverse semigroup S is left absolutely flat if and only if S is a right generalized inverse semigroup and the (nonidentity) structure maps of E(S) are constant. In particular all inverse semigroups are left (and right) absolutely flat (see [1]). Other consequences are derived.
Algebra Universalis, Dec 1, 1972
Mathematische Nachrichten, Dec 1, 2005
Page 1. Math. Nachr. 278, No. 15, 1743 1755 (2005) / DOI 10.1002/mana.200310338 Lazard's T... more Page 1. Math. Nachr. 278, No. 15, 1743 1755 (2005) / DOI 10.1002/mana.200310338 Lazard's Theorem for S-posets Sydney BulmanFleming ∗1 and Valdis Laan2 1 Wilfrid Laurier University, Waterloo, Canada N2L 3C5 2 University of Tartu, Tartu, Estonia ...
Semigroup Forum, Feb 1, 2004
In Comm. Algebra 30 (3) (2002), 1475-1498, Bulman-Fleming and Kilp developed various notions of f... more In Comm. Algebra 30 (3) (2002), 1475-1498, Bulman-Fleming and Kilp developed various notions of flatness of a right act A S over a monoid S that are based on the extent to which the functor A S ⊗ − preserves equalizers. In Semigroup Forum 65 (3) (2002), 428-449, Bulman-Fleming discussed in detail one of these notions, annihilator-flatness. The present paper is devoted to two more of these notions, weak equalizer-flatness and strong torsion-freeness. An act A S is called weakly equalizer-flat if the functor A S ⊗ − 'almost' preserves equalizers of any two homomorphisms into the left act S S, and strongly torsionfree if this functor 'almost' preserves equalizers of any two homomorphisms from S S into the Rees factor act S (S/Sc), where c is any right cancellable element of S. (The adverb 'almost' signifies that the canonical morphism provided by the universal property of equalizers may be only a monomorphism rather than an isomorphism.) From the definitions it is clear that flatness implies weak equalizer-flatness, which in turn implies annihilator-flatness, and it was known already that both of these implications are strict. A monoid is called right absolutely weakly equalizer-flat if all of its right acts are weakly equalizerflat. In this paper we prove a result which implies that right PP monoids with central idempotents are absolutely weakly equalizer-flat. We also show that for a relatively large class of commutative monoids, right absolute equalizer-flatness and right absolute annihilator-flatness coincide. Finally, we provide examples showing that the implication between strong torsion-freeness and torsion-freeness is strict.