{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/91175"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/91175","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Homomorphisms of wn-right cancellative, wn-bisimple, and wnI-bisimple semigroups","abstract":"R. J. Warne has defined an wn-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (Io)n, where Io is the set of non-negative integers and n is a natural number, under the reverse lexicographic order. Warne has described, modulo groups, the structure of such semigroups [\"Bisimple Inverse Semigroups Mod Groups,\" Duke Math. J., Vol. 14 (1967), pp. 787-811]. He has used this structure and the theory of right cancellative semigroups having identity on which Green's relation J: is a congruence to describe the homomorphisms of an ωn-right cancellative semigroup into an ωn-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n [\"Lectures in Semigroups,\" West Virginia Univ., unpublished]. We have described, modulo groups, the homomorphisms of an ωn-right cancellative semigroup into an ωm-right cancellative semigroup for arbitrary natural numbers n and m. One of the main results is the following: Theorem: Let P = (G ,(Io)n , γ₁,...,γn, w₁,…,wØ(n)) and P* = (G ,(Io)n , α₁,...,αn, t₁,…,tØ(n)) be ωn-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,zn be elements of G* and let f be a homomorphism of G into G* such that (1) (Af)(Ukg)Czk = (Aγkf) for A ∈ G where 1 ≤ k ≤ n and (2) ((zk+s)(Ukg)(Ukg)(Uk+sg)Czk = wØ(n-k)+sf where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements Uk (1 ≤ k ≤ n) are generators of (Io)n, xCzk = zkxzk⁻¹ for x ∈ G*, and xa,ab in G* (x ∈ G*; a,b ∈ (Io)n are specified. Define, for (A,a₁,...,an) ∈ P, (A,a₁,...,an)M = [(Af)(a₁,...,an)h,(a₁,...,an)g] where h is a specified function from (Io)n into G* and g is a determined endomorphism of (Io)n. Then, M is a homomorphism of P into P* and every homomorphism of P into P* is obtained in this fashion. M is an isomorphism if and only if f and g are isomorphisms. M is onto when g is the identity and f is onto. Results similar to this theorem have been obtained when P* is an ωm-right cancellative semigroup with m < n and m > n. Let I be the set of integers. Let S be a bisimple semigroup and let ES denote the set of idempotents of S. S is called ωn-bisimple if and only if ES, under its natural order, is order isomorphic to I x (Io)n under the reverse lexicographic order n ≥ 1. S is called I-bisimple if and only if ES, under its natural order, is order isomorphic to I under the reverse usual order. Warne has described, modulo groups, the structure of ωn-bisimple, ωnI-bisirnple and I-bisimple semigroups in [\"Bisimple Inverse Semigroups Mod Groups,\" Duke Math. J., Vol. 14 (1967), pp. 787-811], [\"ωnI-bisimple Semigroups,\" to appear], and [\"I-bisimple Semigroups,\" Trans. Amer. Math. Soc., Vol. 130 (1968), pp. 367-386] respectively. We have described the homomorphisms of S into S* , by use of the homomorphism theory of ωn-right cancellative semigroups, for the cases (i) S ωn-bisimple and S* ωm-bisimple and (ii) S I-bisimple or ωnI-bisimple and S* I-bisimple or ωmI-bisimple where m and n are natural numbers. The homomorphisms of S onto S* are specified for cases (i) and (ii). Warne has determined the homomorphisms of S onto S* in certain of these cases as he studied the extensions and the congruences of ωn-bisimple, ωnI-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date.","abstract_html":"R. J. Warne has defined an w<sup>n</sup>-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (I<sup>o</sup>)<sup>n</sup>, where I<sup>o</sup> is the set of non-negative integers and n is a natural number, under the reverse lexicographic order. Warne has described, modulo groups, the structure of such semigroups ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811]. He has used this structure and the theory of right cancellative semigroups having identity on which Green's relation J: is a congruence to describe the homomorphisms of an ω<sup>n</sup>-right cancellative semigroup into an ω<sup>n</sup>-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n ["Lectures in Semigroups," West Virginia Univ., unpublished]. We have described, modulo groups, the homomorphisms of an ω<sup>n</sup>-right cancellative semigroup into an ω<sup>m</sup>-right cancellative semigroup for arbitrary natural numbers n and m. One of the main results is the following: Theorem: Let P = (G ,(I<sup>o</sup>)<sup>n</sup> , γ₁,...,γ<sub>n</sub>, w₁,…,w<sub>Ø(n)</sub>) and P<sup>*</sup> = (G ,(I<sup>o</sup>)<sup>n</sup> , α₁,...,α<sub>n</sub>, t₁,…,t<sub>Ø(n)</sub>) be ω<sup>n</sup>-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,z<sub>n</sub> be elements of G<sup>*</sup> and let f be a homomorphism of G into G<sup>*</sup> such that (1) (Af)<sup>(U<sub>k</sub>g)</sup>C<sub>z<sub>k</sub></sub> = (Aγ<sub>k</sub>f) for A ∈ G where 1 ≤ k ≤ n and (2) ((z<sub>k+s</sub>)<sup>(U<sub>k</sub>g)</sup>(U<sub>k</sub>g)<sup>(U<sub>k+s</sub>g)</sup>C<sub>z<sub>k</sub></sub> = w<sub>Ø(n-k)+s</sub>f where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements U<sub>k</sub> (1 ≤ k ≤ n) are generators of (I<sup>o</sup>)<sup>n</sup>, xC<sub>z<sub>k</sub></sub> = z<sub>k</sub>xz<sub>k</sub>⁻¹ for x ∈ G<sup>*</sup>, and x<sup>a</sup>,a<sup>b</sup> in G<sup>*</sup> (x ∈ G<sup>*</sup>; a,b ∈ (I<sup>o</sup>)<sup>n</sup> are specified. Define, for (A,a₁,...,a<sub>n</sub>) ∈ P, (A,a₁,...,a<sub>n</sub>)M = [(Af)(a₁,...,a<sub>n</sub>)h,(a₁,...,a<sub>n</sub>)g] where h is a specified function from (I<sup>o</sup>)<sup>n</sup> into G* and g is a determined endomorphism of (I<sup>o</sup>)<sup>n</sup>. Then, M is a homomorphism of P into P* and every homomorphism of P into P* is obtained in this fashion. M is an isomorphism if and only if f and g are isomorphisms. M is onto when g is the identity and f is onto. Results similar to this theorem have been obtained when P* is an ω<sup>m</sup>-right cancellative semigroup with m < n and m > n. Let I be the set of integers. Let S be a bisimple semigroup and let E<sub>S</sub> denote the set of idempotents of S. S is called ω<sup>n</sup>-bisimple if and only if E<sub>S</sub>, under its natural order, is order isomorphic to I x (I<sup>o</sup>)<sup>n</sup> under the reverse lexicographic order n ≥ 1. S is called I-bisimple if and only if E<sub>S</sub>, under its natural order, is order isomorphic to I under the reverse usual order. Warne has described, modulo groups, the structure of ω<sup>n</sup>-bisimple, ω<sup>n</sup>I-bisirnple and I-bisimple semigroups in ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811], ["ω<sup>n</sup>I-bisimple Semigroups," to appear], and ["I-bisimple Semigroups," Trans. Amer. Math. Soc., Vol. 130 (1968), pp. 367-386] respectively. We have described the homomorphisms of S into S* , by use of the homomorphism theory of ω<sup>n</sup>-right cancellative semigroups, for the cases (i) S ω<sup>n</sup>-bisimple and S* ω<sup>m</sup>-bisimple and (ii) S I-bisimple or ω<sup>n</sup>I-bisimple and S* I-bisimple or ω<sup>m</sup>I-bisimple where m and n are natural numbers. The homomorphisms of S onto S* are specified for cases (i) and (ii). Warne has determined the homomorphisms of S onto S* in certain of these cases as he studied the extensions and the congruences of ω<sup>n</sup>-bisimple, ω<sup>n</sup>I-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date.","abstract_has_math":false,"creators":["Hogan, John Wesley"],"institution":"Virginia Polytechnic Institute","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1969,"date_issued":"1969","date_published":"1969","updated_at":"2026-07-22T22:19:07Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/91175","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Hogan, John Wesley"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-07-03T20:52:14Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-07-03T20:52:14Z"]},{"key":"dc:date.issued","label":"Date","values":["1969"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/91175"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["R. J. Warne has defined an wn-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (Io)n, where Io is the set of non-negative integers and n is a natural number, under the reverse lexicographic order. Warne has described, modulo groups, the structure of such semigroups [\"Bisimple Inverse Semigroups Mod Groups,\" Duke Math. J., Vol. 14 (1967), pp. 787-811]. He has used this structure and the theory of right cancellative semigroups having identity on which Green's relation J: is a congruence to describe the homomorphisms of an ωn-right cancellative semigroup into an ωn-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n [\"Lectures in Semigroups,\" West Virginia Univ., unpublished]. We have described, modulo groups, the homomorphisms of an ωn-right cancellative semigroup into an ωm-right cancellative semigroup for arbitrary natural numbers n and m. One of the main results is the following: Theorem: Let P = (G ,(Io)n , γ₁,...,γn, w₁,…,wØ(n)) and P* = (G ,(Io)n , α₁,...,αn, t₁,…,tØ(n)) be ωn-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,zn be elements of G* and let f be a homomorphism of G into G* such that (1) (Af)(Ukg)Czk = (Aγkf) for A ∈ G where 1 ≤ k ≤ n and (2) ((zk+s)(Ukg)(Ukg)(Uk+sg)Czk = wØ(n-k)+sf where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements Uk (1 ≤ k ≤ n) are generators of (Io)n, xCzk = zkxzk⁻¹ for x ∈ G*, and xa,ab in G* (x ∈ G*; a,b ∈ (Io)n are specified. Define, for (A,a₁,...,an) ∈ P, (A,a₁,...,an)M = [(Af)(a₁,...,an)h,(a₁,...,an)g] where h is a specified function from (Io)n into G* and g is a determined endomorphism of (Io)n. Then, M is a homomorphism of P into P* and every homomorphism of P into P* is obtained in this fashion. M is an isomorphism if and only if f and g are isomorphisms. M is onto when g is the identity and f is onto. Results similar to this theorem have been obtained when P* is an ωm-right cancellative semigroup with m < n and m > n. Let I be the set of integers. Let S be a bisimple semigroup and let ES denote the set of idempotents of S. S is called ωn-bisimple if and only if ES, under its natural order, is order isomorphic to I x (Io)n under the reverse lexicographic order n ≥ 1. S is called I-bisimple if and only if ES, under its natural order, is order isomorphic to I under the reverse usual order. Warne has described, modulo groups, the structure of ωn-bisimple, ωnI-bisirnple and I-bisimple semigroups in [\"Bisimple Inverse Semigroups Mod Groups,\" Duke Math. J., Vol. 14 (1967), pp. 787-811], [\"ωnI-bisimple Semigroups,\" to appear], and [\"I-bisimple Semigroups,\" Trans. Amer. Math. Soc., Vol. 130 (1968), pp. 367-386] respectively. We have described the homomorphisms of S into S* , by use of the homomorphism theory of ωn-right cancellative semigroups, for the cases (i) S ωn-bisimple and S* ωm-bisimple and (ii) S I-bisimple or ωnI-bisimple and S* I-bisimple or ωmI-bisimple where m and n are natural numbers. The homomorphisms of S onto S* are specified for cases (i) and (ii). Warne has determined the homomorphisms of S onto S* in certain of these cases as he studied the extensions and the congruences of ωn-bisimple, ωnI-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Homomorphisms of wn-right cancellative, wn-bisimple, and wnI-bisimple semigroups"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics"],"dc:creator":["Hogan, John Wesley"],"dc:date.accessioned":["2019-07-03T20:52:14Z"],"dc:date.available":["2019-07-03T20:52:14Z"],"dc:date.issued":["1969"],"dc:description.abstract":["R. J. Warne has defined an wn-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (Io)n, where Io is the set of non-negative integers and n is a natural number, under the reverse lexicographic order. Warne has described, modulo groups, the structure of such semigroups [\"Bisimple Inverse Semigroups Mod Groups,\" Duke Math. J., Vol. 14 (1967), pp. 787-811]. He has used this structure and the theory of right cancellative semigroups having identity on which Green's relation J: is a congruence to describe the homomorphisms of an ωn-right cancellative semigroup into an ωn-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n [\"Lectures in Semigroups,\" West Virginia Univ., unpublished]. We have described, modulo groups, the homomorphisms of an ωn-right cancellative semigroup into an ωm-right cancellative semigroup for arbitrary natural numbers n and m. One of the main results is the following: Theorem: Let P = (G ,(Io)n , γ₁,...,γn, w₁,…,wØ(n)) and P* = (G ,(Io)n , α₁,...,αn, t₁,…,tØ(n)) be ωn-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,zn be elements of G* and let f be a homomorphism of G into G* such that (1) (Af)(Ukg)Czk = (Aγkf) for A ∈ G where 1 ≤ k ≤ n and (2) ((zk+s)(Ukg)(Ukg)(Uk+sg)Czk = wØ(n-k)+sf where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements Uk (1 ≤ k ≤ n) are generators of (Io)n, xCzk = zkxzk⁻¹ for x ∈ G*, and xa,ab in G* (x ∈ G*; a,b ∈ (Io)n are specified. Define, for (A,a₁,...,an) ∈ P, (A,a₁,...,an)M = [(Af)(a₁,...,an)h,(a₁,...,an)g] where h is a specified function from (Io)n into G* and g is a determined endomorphism of (Io)n. Then, M is a homomorphism of P into P* and every homomorphism of P into P* is obtained in this fashion. M is an isomorphism if and only if f and g are isomorphisms. M is onto when g is the identity and f is onto. Results similar to this theorem have been obtained when P* is an ωm-right cancellative semigroup with m < n and m > n. Let I be the set of integers. Let S be a bisimple semigroup and let ES denote the set of idempotents of S. S is called ωn-bisimple if and only if ES, under its natural order, is order isomorphic to I x (Io)n under the reverse lexicographic order n ≥ 1. S is called I-bisimple if and only if ES, under its natural order, is order isomorphic to I under the reverse usual order. Warne has described, modulo groups, the structure of ωn-bisimple, ωnI-bisirnple and I-bisimple semigroups in [\"Bisimple Inverse Semigroups Mod Groups,\" Duke Math. J., Vol. 14 (1967), pp. 787-811], [\"ωnI-bisimple Semigroups,\" to appear], and [\"I-bisimple Semigroups,\" Trans. Amer. Math. Soc., Vol. 130 (1968), pp. 367-386] respectively. We have described the homomorphisms of S into S* , by use of the homomorphism theory of ωn-right cancellative semigroups, for the cases (i) S ωn-bisimple and S* ωm-bisimple and (ii) S I-bisimple or ωnI-bisimple and S* I-bisimple or ωmI-bisimple where m and n are natural numbers. The homomorphisms of S onto S* are specified for cases (i) and (ii). Warne has determined the homomorphisms of S onto S* in certain of these cases as he studied the extensions and the congruences of ωn-bisimple, ωnI-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/91175"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Homomorphisms of wn-right cancellative, wn-bisimple, and wnI-bisimple semigroups"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute"]},"updated_at":"2026-07-22T22:19:07Z"}