Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics
This paper deals with a broad question—to what extent is topology algebraic—using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the a...
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| Language: | English |
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2007-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2007/43645 |
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| author | S. E. Rodabaugh |
| author_facet | S. E. Rodabaugh |
| author_sort | S. E. Rodabaugh |
| collection | DOAJ |
| description | This paper deals with a broad question—to what extent is topology
algebraic—using two specific questions: (1) what are the algebraic
conditions on the underlying membership lattices which insure that
categories for topology and fuzzy topology are indeed topological
categories; and (2) what are the algebraic conditions which insure that
algebraic theories in the sense of Manes are a foundation for the
powerset theories generating topological categories for topology and fuzzy
topology? This paper answers the first question by generalizing the Höhle-Šostak
foundations for fixed-basis lattice-valued topology and the Rodabaugh foundations for variable-basis lattice-valued topology using semi-quantales; and it answers the second question by giving
necessary and sufficient conditions under which certain theories—the very
ones generating powerset theories generating (fuzzy) topological theories in
the sense of this paper—are algebraic theories, and these conditions use
unital quantales. The algebraic conditions answering the second question are
much stronger than those answering the first question. The syntactic
benefits of having an algebraic theory as a foundation for the powerset
theory underlying a (fuzzy) topological theory are explored; the
relationship between these two specific questions is discussed; the role of
pseudo-adjoints is identified in variable-basis powerset theories which are
algebraically generated; the relationships between topological theories in
the sense of Adámek-Herrlich-Strecker and topological theories in the sense
of this paper are fully resolved; lower-image operators introduced for fixed-basis
mathematics are completely described in terms of standard image operators;
certain algebraic theories are given which determine powerset theories
determining a new class of variable-basis categories for topology and fuzzy
topology using new preimage operators; and the theories of this paper are
undergirded throughout by several extensive inventories of examples. |
| format | Article |
| id | doaj-art-2f9f993ad1094d94b2ea086213cc9d57 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-2f9f993ad1094d94b2ea086213cc9d572025-02-03T05:45:23ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252007-01-01200710.1155/2007/4364543645Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued MathematicsS. E. Rodabaugh0Department of Mathematics and Statistics, Youngstown State University, Youngstown, OH 44555-3609, USAThis paper deals with a broad question—to what extent is topology algebraic—using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the algebraic conditions which insure that algebraic theories in the sense of Manes are a foundation for the powerset theories generating topological categories for topology and fuzzy topology? This paper answers the first question by generalizing the Höhle-Šostak foundations for fixed-basis lattice-valued topology and the Rodabaugh foundations for variable-basis lattice-valued topology using semi-quantales; and it answers the second question by giving necessary and sufficient conditions under which certain theories—the very ones generating powerset theories generating (fuzzy) topological theories in the sense of this paper—are algebraic theories, and these conditions use unital quantales. The algebraic conditions answering the second question are much stronger than those answering the first question. The syntactic benefits of having an algebraic theory as a foundation for the powerset theory underlying a (fuzzy) topological theory are explored; the relationship between these two specific questions is discussed; the role of pseudo-adjoints is identified in variable-basis powerset theories which are algebraically generated; the relationships between topological theories in the sense of Adámek-Herrlich-Strecker and topological theories in the sense of this paper are fully resolved; lower-image operators introduced for fixed-basis mathematics are completely described in terms of standard image operators; certain algebraic theories are given which determine powerset theories determining a new class of variable-basis categories for topology and fuzzy topology using new preimage operators; and the theories of this paper are undergirded throughout by several extensive inventories of examples.http://dx.doi.org/10.1155/2007/43645 |
| spellingShingle | S. E. Rodabaugh Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics International Journal of Mathematics and Mathematical Sciences |
| title | Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics |
| title_full | Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics |
| title_fullStr | Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics |
| title_full_unstemmed | Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics |
| title_short | Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics |
| title_sort | relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice valued mathematics |
| url | http://dx.doi.org/10.1155/2007/43645 |
| work_keys_str_mv | AT serodabaugh relationshipofalgebraictheoriestopowersettheoriesandfuzzytopologicaltheoriesforlatticevaluedmathematics |