Identity Congruence Research Articles

Properties of relatively free inverse semigroups

The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if S S is combinatorial (i.e., H \mathcal {H} is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, S S does not contain a copy of the bicyclic semigroup) or E E -unitary (i.e., E ( S ) E(S) is the kernel of the minimum group congruence) then the relatively free inverse semigroup F V X F{\mathcal {V}_X} on the set X X in the variety V \mathcal {V} generated by S S is also combinatorial, completely semisimple or E E -unitary, respectively. If S S is a fundamental (i.e., the only congruence contained in H \mathcal {H} is the identity congruence) and | X | ⩾ ℵ 0 |X| \geqslant {\aleph _0} , then F V X F{\mathcal {V}_X} is also fundamental. F V X F{\mathcal {V}_X} may not be fundamental if | X | > ℵ 0 |X| > {\aleph _0} . It is also shown that for any variety of groups U \mathcal {U} and for | X | ⩾ ℵ 0 |X| \geqslant {\aleph _0} , there exists a variety of inverse semigroups V \mathcal {V} which is minimal with respect to the properties (i) F V X F{\mathcal {V}_X} is fundamental and (ii) V ∩ G = U \mathcal {V} \cap \mathcal {G} = \mathcal {U} , where G \mathcal {G} is the variety of groups. In the main result of the paper it is shown that there exists a variety V \mathcal {V} for which F V X F{\mathcal {V}_X} is not completely semisimple, thereby refuting a long standing conjecture.

Eventually regular semigroups

A semigroup is called eventually regular if each of its elements hassome power that is regular. Thus the class of all eventually regular semi-groups includes both the class of all regular semigroups and the class ofall group-bound semigroups and so in particular includes the class of allfinite semigroups [J].Many results that hold for all regular semigroups also hold for allfinite semigroups; often this occurrence is not just a coincidence but isnecessarily the case since the results concerned hold for eventuallyregular semigroups. We show that many results may be generalized fromregular semigroups to eventually regular semigroups. In particularLallement's Lemma that for every congruence p on a regular semigroup 5 ,every idempotent p-class contains an idempotent is shown to hold foreventually regular semigroups [I].We define a relation that we denote by \i = u(S) on an arbitrarysemigroup S and show that u is an idempotent-separating congruence onS . For an eventually regular semigroup 5 it is shown that \i is themaximum idempotent-separating congruence on S . Let S be an arbitrarysemigroup. We show that the semigroup S/p is finite if and only if theset of idempotents of S , E(S) is finite [1]. The semigroup S iscalled fundamental if the only idempotent-separating congruence on S isu . It is shown that p is the identity congruence on S/]i(S) [2].(Using the previous result David Easdown has shown that for any semigroupReceived 16 Augus 198t U . Thesis submitted to Monash University February198U. Degree approved August 198U. Supervisor: Professor G.B. Preston.Copyright Clearance Centre, Inc. Serial-fee code: OOOU-9727/85$A2.00 + 0.00.157

Fundamental semigroups

SynopsisAn idempotent-separating congruence μ is studied further in this paper. It is shown to satisfy special properties with respect to regular elements and to group-bound elements. It is shown that for any semigroup S, μ is the identity congruence on S/μ. From this, it can be shown that S/μ is fundamental for any semigroup S. Some alternative characterizations of μ are given and applied to yield sufficient conditions for a subsemigroup T of S to satisfy μ (T) = μ (S) ∩ (T × T), whence T is fundamental if S is fundamental.

Gay fathers: Evolution of the gay-father identity.

Interview data from an exploratory study of 18 gay fathers indicate that identity congruence evolves over time as these men participate in both the world of fathers and the world of gays. Disclosure of each identity in the opposite world, and the acceptance of both identities by intimate others, are also crucial to the gay father's achievement of self-acceptance.

Identity, Militancy, and Cultural Congruence: The Menominee and Kainai

Recent movements of varying degrees of mili tancy on the part of American Indians can be better under stood if we have a grasp of the kinds of adaptations Indians have made to the long-term and continuing confrontation with white culture, white power, and white world views. Native Americans do not constitute a single group. The Menominee are taken as an example of an Indian tribe with a hitherto unaggressive record that has recently engaged in militant activity. The diversity within the Menominee popula tion is described in terms of four major types of long-standing adaptation that were observed as dominant in the 1950s and 1960s and that emerged some time before that. Recent militancy is regarded as a fifth type of adaptive response to the continuing confrontation between Menominee and white culture. Militancy is interpreted, in part, as an assertion of identity. The responses of the Kainai, the Blood Indians of Alberta, Canada, to white culture and power are contrasted briefly at certain critical points to demonstrate the fact of diversity among American Indians in regard to current actions and to reinforce the interpretation that the degree of difference in cultures and world views between Indian cultures and white culture is a significant factor in the kinds of adaptive response to confrontation native American groups have made and will make. his doctorate at the University of California at Los Angeles in 1952. They have done much of their field work together, in three American Indian communities, and most recently, in Germany, where over the past decade they have studied the influence of elementary schools on the adaptations children make to an urbanizing environment. Together they have published a number of books and articles, including one on the Menominee published in 1971, Dreamers Without Power. The Spindlers were editors of the AMERICAN ANTHROPOLOGIST from 1963 to 1967. 1. Our terminology will vacillate between American Indians and "Native Americans," as it does in the current literature. 2. The term "Whites". or "Whiteman" is used by many Indians to refer to all non- Indians in a general way meaning people who live in the United States in a main stream, Anglo manner. We will use the term "white" as an adjective and "White" as a noun, as a convenience, to avoid cumber some alternative terminology. 3. See George D. Spindler and Louise S. Spindler, "Fieldwork among the Me nomini," in Being an Anthropologist: Field work in Eleven Cultures, ed. George D. Spindler (New York: Holt, Rinehart and Winston, 1970), pp. 267-302. 4. See George D. Spindler, ed., The Mak ing of Psychological Anthropology (Berkeley: University of California Press, 1978) for an analysis of the history of the field and a chapter by Louise S. Spindler on the re search with the Menominee and Blood. 5. See Louise S. Spindler, "The Me nominee," in Handbook of North Ameri can Indians (Washington, D.C.: Smithsonian Institution, forthcoming) for a synopsis of the history and ethnohistory of the Menominee. 6. Our understanding of the traditional Me nominee culture and the confrontation with Western culture is developed in Dreamers Without Power: The Menomini Indians (New York: Holt, Rinehart and Winston, 1971). 7. See George D. Spindler and Louise S. Spindler, "The Instrumental Activities In ventory : A Technique for the Study of the Psychology of Acculturation," Southwestern Journal of Anthropology, vol. 21 (1965), pp. 1-23 for a synopsis of Blood cognitive orien tations. 8. The chapter on the peyote road in Dreamers Without Power combines the ob servations of the Spindlers and of J. Sidney Slotkin, who made a special study of the Na tive American Church ( The Peyote Religion: A Study in Indian-White Relations [Glencoe, Ill.: The Free Press, 1956] and Menomini Peyotism [Philadelphia, Pa.: American Philosophical Society, 1952]). 9. Others have done research with the Menominee during the past two decades, and we cannot catalogue their influence upon our thinking except to acknowledge that of William H. Hodge, in private conversa tions. Our interpretations differ, in part due to the different time periods in which we have worked, in part due to his focus on urban Menominee, and our focus on the native- oriented and peyote groups, and in part due to differences in our world views. His "Eth nicity as a Factor in Modern American Indian Migration: A Winnebago Case Study with References to Other Indian Situations," in Migration and Development, eds. H. I. Safa and B. M. Dutoit (The Hague, Mouton, 1975) has proven useful. Neils W. Braroe, Indian and White: Self-Image and Interaction in a Canadian Plains Community (Stanford, Calif.: Stanford University Press, 1975) has also in fluenced our thinking about Indian-white relations.

Congruence-free inverse semigroups with zero

A semigroup is said to be congruence-free if it has only two congruences, the identity congruence and the universal congruence. It is almost immediate that a congruence-free semigroup of order greater than two must either be simple or 0-simple. In this paper we describe the semilattices of congruence-free inverse semi-groups with zero. Further, congruence-free inverse semigroups with zero are characterized in terms of partial isomorphisms of their semilattices. A general discussion of congruence-free inverse semigroups, with and without zero, is given by Munn (to appear).