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This work was partially supported by the Portuguese Foundation for Science and Technology through the grant UID/MAT/00297/2013 (CMA). Extending the notions of inverse transversal and associate subgroup, we consider a regular semigroup S with the property that there exists a subsemigroup T which contains, for each x∈S, a unique y such that both xy and yx are idempotent. Such a subsemigroup is necessarily a group which we call a special subgroup. Here we investigate regular semigroups with this property. In particular, we determine when the subset of perfect elements is a subsemigroup and describe its structure in naturally arising situations. Postprint

An ordered regular semigroup S is E-special if for every x ∈ S there is a biggest x + ∈ S such that both xx + and x + x are idempotent. Every regular strong Dubreil–Jacotin semigroup is E-special, as is every ordered completely simple semigroup with biggest inverses. In an E-special ordered regular semigroup S in which the unary operation x → x + is antitone the subset P of perfect elements is a regular ideal, the biggest inverses in which form an inverse transversal of P if and only if S has a biggest idempotent. If S + is a subsemigroup and S does not have a biggest idempotent, then P contains a copy of the crown bootlace semigroup.

On a regular semigroup S a compatible order ≤is defined to be ℋ-cohesive if elements that are comparable under ≤are ℋ-equivalent. In the case where S admits an inverse transversal a complete description of the ℋ-cohesive orders on S is obtained. When S is locally inverse the semilattices of ℋ-cohesive orders and amenable orders are isomorphic, and coincide if and only if S is completely simple.

A simpler-than-hitherto description is obtained of all amenable orders definable on a regular semigroup with a quasi-ideal (in particular, a multiplicative) inverse transversal.

We first consider an ordered regular semigroup S in which every element has a biggest inverse and determine necessary and sufficient conditions for the subset S ○ of biggest inverses to be an inverse transversal of S. Such an inverse transversal is necessarily weakly multiplicative. We then investigate principally ordered regular semigroups S with the property that S ○ is an inverse transversal. In such a semigroup we determine precisely when the set S ☆ of biggest pre-inverses is a subsemigroup and show that in this case S ☆ is itself an inverse transversal of a subsemigroup of S. The ordered regular semigroup of 2 × 2 boolean matrices provides an informative illustrative example. The structure of S, when S ☆ is a group, is also described.

An endomorphism on an algebra 𝒜 is said to be “strong” if it is compatible with every congruence on 𝒜; and 𝒜 is said to have the “strong endomorphism kernel property” if every congruence on 𝒜, different from the universal congruence, is the kernel of a strong endomorphism on 𝒜. Here we consider this property in the context of Ockham algebras. In particular, for those MS-algebras that have this property we describe the structure of their dual space in terms of 1-point compactifications of discrete spaces.

An algebra 𝒜 has the endomorphism kernel property if every congruence on 𝒜 different from the universal congruence is the kernel of an endomorphism on 𝒜. We first consider this property when 𝒜 is a finite distributive lattice, and show that it holds if and only if 𝒜 is a cartesian product of chains. We then consider the case where 𝒜 is an Ockham algebra, and describe in particular the structure of the finite de Morgan algebras that have this property.

Here we investigate the structure of the lattice of congruences of a subdirectly irreducible extended Ockham algebra. In particular, we determine precisely when it is a chain.

We consider an ordered completely simple semigroup S = M(G; I, Λ; P) which is non-degenerate in the sense that G, I, Λ are non-trivial. We prove that if every element of S has a biggest inverse then S contains at least one of seven particular types of subsemigroup.

A natural characterization of known types of inverse transversal leads to the introduction of several new types, which we characterize in a similar natural way. Relations between the various types produce a general classification. We also determine for which types of inverse transversal considered the congruence extension property holds.

It has been suspected for some time that if a regular semigroup S has an inverse transversal then all inverse transversals of S are isomorphic. Here we give a proof of this fact.

Here we initiate an investigation into the class eO of extended Ockham algebras, namely bounded distributive lattices endowed with a commuting pair of unary operations, one of which is an endomorphism and the other is a dual endomorphism. A consideration of the subdirectly irreducible algebras leads, via Priestley duality, to a description of all finite subdirectly irreducible eO-algebras. In particular, we show that there are precisely nine non-isomorphic subdirectly irreducible symmetric extended de Morgan algebras, all of which are simple. We also describe the corresponding lattice of varieties.

The variety dpk 1,1 consists of those algebras (L;⋁,⋀,f,*,0,1) of type (2,2,1,1,0,0) where (L;⋁,⋀,f,0,1) is an Ockham algebra in which f 3 = f, (L;⋁,⋀,*,0,1) is a demi-pseudocomplemented lattice, and the unary operations f and * commute. Here we give a description of the structure of the lattice of congruences of a subdirectly irreducible algebra in dpk 1,1.

The variety pOconsists of those algebras (L;⋁,⋀,f,*,0,1) where (L;⋁,⋀,f,0,1) is an Ockham algebra, (L;⋁,⋀,f,*,0,1) is a p-algebra, and the unary operations fand *. commute. For an algebra in pK ωwe show that the compact congruences form a dual Stone lattice and use this to determine necessary and sufficient conditions for a principal congruence to be complemented. We also describe the lattice of subvarieties of pK 1,1identifying therein the biggest subvariety in which every principal congruence is complemented, and the biggest subvariety in which the intersection of two principal congruences is principal.

The variety pO consists of those algebras (L; ⋀, ⋁, f,*, 0,1) of type (2,2,1,1,0,0) where (L; ⋀, ⋁, f, 0,1) is an Ockham algebra, (Z; ⋀, ⋁,*, 0,1) is a p-algebra, and the unary operations f and * commute. We describe completely the structure of the subdirectly irreducible algebras that belong to the subclass pK1,1, characterised by the property f3 = f.

We describe the structure of a regular semigroup with an associate subgroup the identity element of which is a medial idempotent. As a particular application of this, we obtain the structure of perfect Dubreil-Jacotin semigroups in which the set of residuals of the bimaximum element form a subgroup., Fundação para a Ciência e Tecnologia (FCT), Fundação Calouste Gulbenkian

Given any family of normal subgroups of a group, we construct in a natural way a certain monoid, the group of units of which is a semidirect product. We apply this to obtain a description of both the semigroup of endomorphisms and the group of automorphisms of an Ockham algebra of finite boolean type. We also determine when such a monoid is regular, orthodox, or inverse

If an Ockham algebra L belongs to a Berman class and its endomorphism semi­group End L is regular then necessarily L ∈ Kp 2 for some p. For a given L∈Kp 2 the question of precisely when End L is regular is solved in the case where L is subdirectly irre­ducible. Using a particular construction, we show that every Berman class Kp 2 contains an algebra L for which End L is an inverse semigroup.

The structure of orthodox semigroups with a normal band of idempotents has been described by Yamada. Since, for a naturally ordered strong Dubreil-Jacotin onhodox semigroup, ir can be shown that the band of idempotents is normal, it is of interest to investigate the structure of these ordered onhodox semigroups via the Yamada decomposition. What is hard in such matters, and no exception here, is how to marry together the order theoretic structure with the algebraic structure theory. Basically, the problem is how to define orders on the building bricks of the structure theory in such a way tha~ the algebraic isomorphisms become order isomorphisms. This we are able to do for a variety of different types, obtaining structure theorems which concern not only cartesian orders but also several new types of lexicographic orders. Examples are given to illustrate the hierarchy of orders and the corresponding algebraic conditions required for the order isomorphisms.

In brief: Football fatality data have been collected since 1931 and catastrophic injury data since 1977. There have been dramatic reductions in both areas, but for the past six years these numbers have remained steady. Some experts in the field of football injury research have stated that further reductions are not possible. The authors argue that efforts should be continued to eliminate these injuries and recommend mandatory medical examinations, rule changes, proper conditioning and equipment, having a physician on the field whenever possible, and continued research to help in this effort.

Improved conditioning, a helmet standard, and rules forbidding butt blocking and face tackling account for the lowest incidence of football head and neck injuries in 18 years.

In brief The authors compiled data from 20 National Collegiate Athletic Association lacrosse teams to study the relationship between injuries, protective equipment, and game rules. Athletic trainers for each team kept track of Injuries as they occurred. The 586 players sustained 306 injuries for an injury rate of 52.2%. Half the injuries occurred in the lower extremities, which have no protective equipment. The trunk and arms received 39.8% of the injuries, and the head and face received 9.8%, even though the players are required to wear helmets and face masks. Fewer than 10% of the athletes wore mouth guards. The authors concluded that lacrosse equipment and rules need further study.

The purpose of this study was to isolate relationships between psychomotor responses and personality structure. Fifteen subjecs were administered the Thurstone Temperament Schedule. Stress was provided by a heat chamber with an ambient temperature of 110 F (30–50 percent R.H.) and walking on a treadmill which moved at 4 mph at zero grade. Analysis of the data permits the following conclusions: (a) two-hand coordination showed a positive significant relationship to the measure of competitiveness; (b) tapping showed a positive significant relationship to the vigorous and sociable traits; and (c) a significant positive relationship between strength and the active trait was found. Hand and arm steadiness had a significant negative relationship with the emotional stability trait.

To determine the effects of water balance on ability to perform in high temperatures, 18 male subjects were subjected to exhaustion runs on the treadmill under three experimental conditions: normal water balance, dehydration, and superhydration. All three runs were performed in an ambient temperature of 120°F. No statistical difference was found between performance under conditions of superhydration and normal water balance, but performance in the dehydrated state was significantly reduced. The performance of a subgroup of 11 athletes under the condition of superhydration was significantly superior to their corresponding performance under the condition of normal water balance. The seven nonathletes did not exhibit this tendency.

To investigate further the inverse relationship that appears to exist between ischemic heart disease and physical activity, experiments were conducted on 54 healthy male students to determine the relationship between blood coagulation time and fat storage, Harvard Step Test scores, age, height, and weight. No significant relationship was found to exist between clotting time and the selected variables.

Eight track athletes were studied to determine the effects of a warm-up and a lack of a warm-up on the heart rate during specified exercise routines. The variables of anticipatory increase in heart rate, maximum heart rate, and recovery decrease in heart rate were studied. The difference between these variables was not significant at the .05 level of confidence. The same variables were correlated with the Harvard step test scores for the subjects. High correlations were found between the variables of maximum heart rate and recovery decrease in heart rate, and the step test scores.

The optimum linear controller for a linear process control system having a limited control power available and subject to a random disturbance is derived and shown to have the same structure as a standard three term controller, Amplitude limitation in the actuator is then introduced and a method is presented for determining the optimum linear controller. With this method the effect of amplitude limitation on optimum controller settings is derived. The effects of varying the disturbance power and the linear regime of the saturating element are calculated and results are confirmed by measurements made on an analogue computer.

(1954). Notes on the Poultry Research Centre Flock of Brown Leghorns. World's Poultry Science Journal: Vol. 10, No. 2, pp. 140-143.