In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class (...) of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Łukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-Łukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free Łukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras are indecomposable, while finitely free generated algebras can be (...) only decomposed into a direct product of two factors, one of which is the two-element implication algebra. (shrink)

AbstractŁukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-Łukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free Łukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras are indecomposable, while finitely free generated algebras can be (...) only decomposed into a direct product of two factors, one of which is the two-element implication algebra. (shrink)

This essay argues the image of the ‘Mystical Body’ or ‘Body of Christ’ is fully intelligible only in light of the Cross. The Body of Christ is a cruciform Body. As the Body of Christ crucified and risen, it is presently being configured to Christ in the world through self-sacrificial love. The essay traces the place of the Cross in some representative twentieth-century Catholic theologies of the Mystical Body, in light of the perspective of Thomas Aquinas. I first survey four (...) theologians from the first half of the century who, in their understandings of the Mystical Body, gave a place to the Cross: Emile Mersch, Fulton Sheen, Charles Journet, and Pius Pasch. Here I also examine Pope Pius XII's encyclical Mystici Corporis. Second, I explore the approach of some notable Thomistic theologians or interpreters of Aquinas from the early 1960s onward: Jerome Hamer, M. J. Le Guillou, George Sabra, Jean-Pierre Torrell, and Herwi Rikhof. In this section, I also examine Vatican II's Lumen Gentium. My third and final section treats Aquinas himself, in order to reflect upon the place of the Cross in his understanding of the Mystical Body, as found in his biblical commentaries and the Summa theologiae. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:484 BOOK REVIEWS faith. Yet faith-knowledge alone is insufficient to account for Jesus' extraordinary gifts as a teacher: for this we must appeal to a special charism along the lines of an infused knowledge. According to Torrell this knowledge is best understood by reference to Aquinas's mature teaching on prophecy: God equipped the prophets with an infused light (but not infused ideas) enabling them to communicate divine truths (...) to others. Likewise, God conferred on Jesus an infused light akin to that of the prophets, but with this qualitative difference: in him the lumen is a permanent feature of his cognitive life (a habitus), while it is given to the prophets only intermittently (per modum actus). The foregoing i;ummary will have served its purpose if it has conveyed to the reader some sense of the very rich historical, philosophical, and theological reflections that comprise this volume. The editorial team of the Revue thomiste is to be commended for the high caliber of this and the other special publications it has produced in recent years: the 1992 commemorative volume on the theological achievement of M.-M. Labourdette; th~ 1993 centenary index ("tables generales 1893-1992"); the Gilson issue of 1994; and most recently an issue devoted to Thomas Aquinas and the onto-theo-logy debate (1995). This reviewer eagerly waits for more. Fordham University Bronx, New York GREGORY M. REICHBERG The Divine Initiative: Grace, World-Order, and Human Freedom in the Early Writings of Bernard Lonergan. By J. MICHAEL STEBBINS. Toronto: University of Toronto Press, 1995. Pp. xxii + 399. $65.00 (cloth). Aquinas insists that the creator's primary intent is the "order of the universe," and Bernard Lonergan's dissertation on "operating grace" in Aquinas's writings managed to move beyond the stalemated discussion of "sufficient" and "efficacious" grace precisely because he displayed how any discourse about grace had to be connected with larger theorems of the creator's operation in creation. What speaking of the divine action called "grace" requires is a set of metaphysical skills adequate to speaking of the "order of the universe" as created. A tall order, whose scope the published edition of his dissertation-Grace and Freedom-so understated that its implications have been missed by many philosophers and theologians fascinated with such questions. Stebbins's careful reconstruction of that text reminds us of its daunting scope. And part of the reason it can do so is that he illustrates both the method and conclusions of Grace and Freedom through a later text which Lonergan had composed (in Latin) for a course on grace offered from 1947 to 1960: De ente supernaturali (which will appear in volume 16 [Early Latin Theology] of the Collected Works of Bernard Lonergan, published by BOOK REVIEWS 485 University of Toronto Press). The significance of this collateral source is that it can profit from and respond to Henri de Lubac's epochal Surnaturel (1946). For those not familiar with Grace and Freedom, as well as some who thought they were, a list of Stebbins's chapters manages to convey the scope of Lonergan's achievement: 1. The Role of Understanding in Theological Speculation; 2. The Principal Instance of Supernatural Being: The Created Communication of the Divine Nature; 3. Thirteenth-Century Breakthrough (The "Theorem" of the Supernatural); 4. Supernatural Transformation of Human Activity; 5. Obediential Potency and the Natural Desire to See God; 6. Molinist and Bannezian Systems; 7. Theoretical Perspective on Divine Concourse; 8. Contingence, Sin and Divine Efficacy. As always, Lonergan must actively reflect on method while executing a theological inquiry, so Stebbins rightly begins with his insistence that such inquiry must be more than "just a networks of concepts; it is primarily an act of understanding" (xix)-Chapter 1. Then, following the expository order of De ente supematurali, he shows how Lonergan roots the supernatural in the theorems developed to speak of the natural: of creation itself. ("Theorem," as we shall see, is a favorite word of Lonergan's, intimating what it takes to move beyond our imaginations to a set of propositions able to articulate the metaphysical issues at stake.) Following the analogy of nature, we can see how the "two operations by which creatures... (shrink)