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Ankylosing spondylitis

Ankylosing spondylitis (AS) is a type of arthritis characterized by long-term inflammation of the joints of the spine, typically where the spine joins the pelvis. With AS, eye and bowel problems—as well as back pain—may occur. Joint mobility in the affected areas sometimes worsens over time. Ankylosing spondylitis is believed to involve a combination of genetic and environmental factors. More than 90% of people affected in the UK have a specific human leukocyte antigen known as the HLA-B27 antigen. The underlying mechanism is believed to be autoimmune or autoinflammatory. Diagnosis is based on symptoms with support from medical imaging and blood tests. AS is a type of seronegative spondyloarthropathy, meaning that tests show no presence of rheumatoid factor (RF) antibodies.

There is no cure for AS. Treatments may include medication, physical therapy, and surgery. Medication therapy focuses on relieving the pain and other symptoms of AS, as well as stopping disease progression by counteracting long-term inflammatory processes. Commonly used medications include NSAIDs, TNF inhibitors, IL-17 antagonists, and DMARDs. Glucocorticoid injections are often used for acute and localized flare-ups.

About 0.1% to 0.8% of the population are affected, with onset typically occurring in young adults. While men and women are equally affected with AS, women are more likely to experience inflammation rather than fusion.

Signs and symptoms
The signs and symptoms of ankylosing spondylitis often appear gradually, with peak onset between 20 and 30 years of age. Initial symptoms are usually a chronic dull pain in the lower back or gluteal region combined with stiffness of the lower back. Individuals often experience pain and stiffness that awakens them in the early morning hours.

As the disease progresses, loss of spinal mobility and chest expansion, with a limitation of anterior flexion, lateral flexion, and extension of the lumbar spine are seen. Systemic features are common with weight loss, fever, or fatigue often present. Pain is often severe at rest but may improve with physical activity. Inflammation and pain may recur to varying degrees regardless of rest and movement.

AS can occur in any part of the spine or the entire spine, often with pain localized to either buttock or the back of the thigh from the sacroiliac joint. Arthritis in the hips and shoulders may also occur. When the condition presents before the age of 18, AS is more likely to cause pain and swelling of large lower limb joints, such as the knees. In prepubescent cases, pain and swelling may also manifest in the ankles and feet where heel pain and enthesopathy commonly develop. Less common occurrences include ectasia of the sacral nerve root sheaths.

About 30% of people with AS will also experience anterior uveitis causing eye pain, redness, and blurred vision. This is thought to be due to the association that both AS and uveitis have with the presence of the HLA-B27 antigen. Cardiovascular involvement may include inflammation of the aorta, aortic valve insufficiency or disturbances of the heart's electrical conduction system. Lung involvement is characterized by progressive fibrosis of the upper portion of the lung.

Pathophysiology
Ankylosing spondylitis (AS) is a systemic rheumatic disease, meaning it affects the entire body. 1–2% of individuals with the HLA-B27 genotype develop the disease. Tumor necrosis factor-alpha (TNF α) and interleukin 1 (IL-1) are also implicated in ankylosing spondylitis. Autoantibodies specific for AS have not been identified. Anti-neutrophil cytoplasmic antibodies (ANCAs) are associated with AS, but do not correlate with disease severity.

Single nucleotide polymorphism (SNP) A/G variant rs10440635 is close to the PTGER4 gene on human chromosome 5 has been associated with an increased number of cases of AS in a population recruited from the United Kingdom, Australia, and Canada. The PTGER4 gene codes for the prostaglandin EP4 receptor, one of four receptors for prostaglandin E2. Activation of EP4 promotes bone remodeling and deposition (see ) and EP4 is highly expressed at vertebral column sites involved in AS. These findings suggest that excessive EP4 activation contributes to pathological bone remodeling and deposition in AS and that the A/G variant rs10440635a of PTGER4 predisposes individuals to this disease, possibly by influencing EP4's production or expression pattern.

The association of AS with HLA-B27 suggests the condition involves CD8 T cells, which interact with HLA-B. This interaction is not proven to involve a self-antigen, and at least in the related reactive arthritis, which follows infections, the antigens involved are likely to be derived from intracellular microorganisms. There is, however, a possibility that CD4+ T lymphocytes are involved in an aberrant way, since HLA-B27 appears to have a number of unusual properties, including possibly an ability to interact with T cell receptors in association with CD4 (usually CD8+ cytotoxic T cell with HLAB antigen as it is a MHC class 1 antigen).

"Bamboo spine" develops when the outer fibers of the fibrous ring (anulus fibrosus disci intervertebralis) of the intervertebral discs ossify, which results in the formation of marginal syndesmophytes between adjoining vertebrae.

Diagnosis
Ankylosing spondylitis is a member of the more broadly defined disease axial spondyloarthritis. Axial spondyloarthritis can be divided into two categories: radiographic axial spondyloarthritis (which is a synonym for ankylosing spondylitis) and non-radiographic axial spondyloarthritis (which include less severe forms and early stages of ankylosing spondylitis).

While AS can be diagnosed through the description of radiological changes in the sacroiliac joints and spine, there are currently no direct tests (blood or imaging) to unambiguously diagnose early forms of ankylosing spondylitis (non-radiographic axial spondyloarthritis). Diagnosis of non-radiologic axial spondyloarthritis is therefore more difficult and is based on the presence of several typical disease features.

These diagnostic criteria include:
 * Inflammatory back pain: Chronic, inflammatory back pain is defined when at least four out of five of the following parameters are present: (1) Age of onset below 40 years old, (2) insidious onset, (3) improvement with exercise, (4) no improvement with rest, and (5) pain at night (with improvement upon getting up). Pain often subsides as the day progresses with movement being of importance to alleviate the joint stiffness.
 * Past history of inflammation in the joints, heels, or tendon-bone attachments
 * Family history for axial spondyloarthritis or other associated rheumatic/autoimmune conditions
 * Positive for the biomarker HLA-B27
 * Good response to treatment with nonsteroidal anti-inflammatory drugs (NSAIDs)
 * Signs of elevated inflammation (C-reactive protein and erythrocyte sedimentation rate)
 * Manifestation of psoriasis, inflammatory bowel disease, or inflammation of the eye (uveitis)

If these criteria still do not give a compelling diagnosis magnetic resonance imaging (MRI) may be useful. MRI can show inflammation of the sacroiliac joint.

X-rays
The earliest changes demonstrable by plain X-ray shows erosions and sclerosis in sacroiliac joints. Progression of the erosions leads to widening of the joint space and bony sclerosis. X-ray spine can reveal squaring of vertebrae with bony spur formation called syndesmophyte. This causes the bamboo spine appearance. A drawback of X-ray diagnosis is the signs and symptoms of AS have usually been established as long as 7–10 years prior to X-ray-evident changes occurring on a plain film X-ray, which means a delay of as long as 10 years before adequate therapies can be introduced.

Options for earlier diagnosis are tomography and MRI of the sacroiliac joints, but the reliability of these tests is still unclear.

Blood parameters
During acute inflammatory periods, people with AS may show an increase in the blood concentration of CRP and an increase in the ESR, but there are many with AS whose CRP and ESR rates do not increase, so normal CRP and ESR results do not always correspond with the amount of inflammation that is actually present. In other words, some people with AS have normal levels of CRP and ESR, despite experiencing a significant amount of inflammation in their bodies.

Genetic testing
Variations of the HLA-B gene increase the risk of developing ankylosing spondylitis, although it is not a diagnostic test. Those with the HLA-B27 variant are at a higher risk than the general population of developing the disorder. HLA-B27, demonstrated in a blood test, can occasionally help with diagnosis, but in itself is not diagnostic of AS in a person with back pain. Over 85% of people that have been diagnosed with AS are HLA-B27 positive, although this ratio varies from population to population (about 50% of African Americans with AS possess HLA-B27 in contrast to the figure of 80% among those with AS who are of Mediterranean descent).

BASDAI
The Bath Ankylosing Spondylitis Disease Activity Index (BASDAI), developed in Bath (UK), is an index designed to detect the inflammatory burden of active disease. The BASDAI can help to establish a diagnosis of AS in the presence of other factors such as HLA-B27 positivity, persistent buttock pain which resolves with exercise, and X-ray or MRI-evident involvement of the sacroiliac joints. It can be easily calculated and accurately assesses the need for additional therapy; a person with AS with a score of four out of a possible 10 points while on adequate NSAID therapy is usually considered a good candidate for biologic therapy.

The Bath Ankylosing Spondylitis Functional Index (BASFI) is a functional index which can accurately assess functional impairment due to the disease, as well as improvements following therapy. The BASFI is not usually used as a diagnostic tool, but rather as a tool to establish a current baseline and subsequent response to therapy.

Children
Juvenile ankylosing spondylitis (JAS) is a rare form of the disease which differs from the more common adult form. Enthesophathy and arthritis of large joints of the lower extremities is more common than the characteristic early-morning back pain seen in adult AS. Ankylosing tarsitis of the ankle is a common feature, as is the more classical findings of seronegative ANA and RF as well as presence of the HLA-B27 allele. Primary engagement of the appendicular joints may explain delayed diagnosis; however, other common symptoms of AS such as uveitis, diarrhea, pulmonary disease and heart valve disease may lead suspicion away from other juvenile spondyloarthropathies.

Schober's test
The Schober's test is a useful clinical measure of flexion of the lumbar spine performed during the physical examination.

Treatment
There is no cure for AS, but treatments and medications can reduce symptoms and pain.

Medication
Medications for AS may be broadly considered either "disease-modifying" or "non-disease-modifying". Disease-modifying medications for ankylosing spondylitis aim to slow disease progression and include drugs like tumor necrosis factor (TNF) inhibitors. Non-disease-modifying medications, such as nonsteroidal anti-inflammatory drugs (NSAIDs), primarily address symptoms like pain and inflammation but do not alter the course of the disease.

NSAIDs
Unless otherwise contraindicated, all people with AS are recommended to take non-steroidal anti-inflammatory drugs (NSAIDs). The dose, frequency, and specific drug may depend on the individual and the symptoms they experience. NSAIDs, such as ibuprofen and naproxen, are used to alleviate pain, reduce inflammation, and improve joint stiffness associated with AS. These medications work by inhibiting the activity of cyclooxygenase (COX) enzymes, which are involved in the production of inflammatory prostaglandins. By reducing the levels of prostaglandins, NSAIDs help mitigate the inflammatory response and relieve symptoms in individuals with ankylosing spondylitis.

TNF inhibitors
Tumor necrosis factor inhibitors (TNFi) are a class of biologic drugs used in the treatment of ankylosing spondylitis. TNFi drugs, such as etanercept, infliximab, adalimumab, certolizumab, and golimumab, target the inflammatory cytokine tumor necrosis factor-alpha (TNF-alpha). TNF-alpha plays a key role in the inflammatory process in ankylosing spondylitis. By blocking TNF-alpha, TNFi drugs help reduce inflammation, pain, and stiffness associated with AS, and may also slow down the progression of spinal damage.

Non-TNFi biologics
Non-TNFi "biologic" drugs used in the treatment of ankylosing spondylitis include drugs that target different pathways involved in the inflammatory process. Two of the most important drugs in this class target IL-17, an important part of the inflammatory system: secukinumab and ixekizumab. They are often considered in cases where TNFi drugs are not effective or cause too many side effects. Additionally, they may sometimes be used as an adjunct to a TNFi when symptoms persist, but improve, while the patient is on the TNFi. The choice of a specific non-TNFi biologic depends on various factors, including the patient's medical history, preferences, and the recommendations of the healthcare provider.

Ustekinumab has frequently been used as a second-line therapy for AS, but it has recently been scrutinized for a lack of efficacy, and is no longer recommended.

Biosimilar drugs
Biosimilar drugs are biological products that are highly similar to an already approved biologic drug, with few or no clinically meaningful differences in terms of safety, purity, and potency. These drugs are developed to be equivalent to the reference biologic, often at a lower cost, providing alternative treatment options. In the context of ankylosing spondylitis, biosimilars are typically used as alternatives to the original biologic drugs. Biosimilars for ankylosing spondylitis may include versions of tumor necrosis factor inhibitors or other biologics commonly used in the treatment of the condition. When possible, physicians are recommended to use the original drugs over the biosimilar versions. Even biosimilars with perfect replication of the quality, composition, and other properties of the original drug are susceptible to nocebo effects.

csARDs
Conventional synthetic antirheumatic drugs (csARDs) are a class of disease-modifying medications. Unlike biologics or targeted synthetic drugs, which act on specific pathways in the immune system, csARDs have a broader effect on the immune system and are often considered traditional or conventional treatments. The most common drugs in this class are methotrexate and sulfasalazine. These medications are only used when others fail, or when certain specific conditions are met, and are often discontinued if a patient's symptoms become manageable with just a TNFi or other medication. Conventional DMARDs such as leflunomide are also considered to be part of this class.

Concerns exist about a possible lack of efficacy of some drugs in this class.

Corticosteroids
Glucocorticoids, such as prednisone or methylprednisolone, are sometimes used in the treatment of ankylosing spondylitis to manage acute flares and provide short-term relief from inflammation and symptoms. They are powerful anti-inflammatory medications that can help reduce pain, swelling, and stiffness associated with AS. However, glucocorticoids are generally not recommended for long-term use. They are more commonly used as localized injections when someone with AS has a temporary pain flare in a particular joint or area.

Surgery
In severe cases of AS, surgery can be an option in the form of joint replacements, particularly in the knees and hips. Surgical correction is also possible for those with severe flexion deformities (severe downward curvature) of the spine, particularly in the neck, although this procedure is considered very risky. In addition, AS can have some manifestations that make anesthesia more complex. Changes in the upper airway can lead to difficulties in intubating the airway, spinal and epidural anesthesia may be difficult owing to calcification of ligaments, and a small number of people have aortic insufficiency. The stiffness of the thoracic ribs results in ventilation being mainly diaphragm-driven, so there may also be a decrease in pulmonary function.

Physical therapy
Though physical therapy remedies have been scarcely documented, some therapeutic exercises are used to help manage lower back, neck, knee, and shoulder pain. There is moderate quality evidence that therapeutic exercise programs help reduce pain and improve function. Therapeutic exercises include:
 * Exercise programs, either at home or supervised
 * Low intensity aerobic exercise, e.g. Pilates
 * Spa-exercise therapy
 * Aquatic physical therapy
 * Proprioceptive neuromuscular facilitation (PNF)
 * Heat therapy
 * Cryotherapy in conjunction with exercise

Diet
Research by Alan Ebringer at King's College in London, beginning in the 1980s, implicates overgrowth of the bacterium Klebsiella pneumoniae in the symptoms of ankylosing spondylitis. The body produces antibodies that attack Klebsiella pneumoniae. Enzymes made by the bacterium resemble human proteins, including three types of collagen (I, III, IV) and the HLA-B27 complex of glycoproteins. The antibodies therefore attack these human proteins, producing the symptoms of ankylosing spondylitis. Ebringer and others recommend low-starch or no-starch diets.

Prognosis
Prognosis is related to disease severity. AS can range from mild to progressively debilitating and from medically controlled to refractory. Some cases may have times of active inflammation followed by times of remission resulting in minimal disability while others never have times of remission and have acute inflammation and pain, leading to significant disability. As the disease progresses, it can cause the vertebrae and the lumbosacral joint to ossify, resulting in the fusion of the spine. This places the spine in a vulnerable state because it becomes one bone, which causes it to lose its range of motion as well as putting it at risk for spinal fractures. This not only limits mobility but reduces the affected person's quality of life. Complete fusion of the spine can lead to a reduced range of motion and increased pain, as well as total joint destruction which could lead to a joint replacement.

Osteoporosis is common in ankylosing spondylitis, both from chronic systemic inflammation and decreased mobility resulting from AS. Over a long-term period, osteopenia or osteoporosis of the AP spine may occur, causing eventual compression fractures and a back "hump". Hyperkyphosis from ankylosing spondylitis can also lead to impairment in mobility and balance, as well as impaired peripheral vision, which increases the risk of falls which can cause fracture of already-fragile vertebrae. Typical signs of progressed AS are the visible formation of syndesmophytes on X-rays and abnormal bone outgrowths similar to osteophytes affecting the spine. In compression fractures of the vertebrae, paresthesia is a complication due to the inflammation of the tissue surrounding nerves.

Organs commonly affected by AS, other than the axial spine and other joints, are the heart, lungs, eyes, colon, and kidneys. Other complications are aortic regurgitation, Achilles tendinitis, AV node block, and amyloidosis. Owing to lung fibrosis, chest X-rays may show apical fibrosis, while pulmonary function testing may reveal a restrictive lung defect. Very rare complications involve neurologic conditions such as the cauda equina syndrome.

Mortality
Mortality is increased in people with AS and circulatory disease is the most frequent cause of death. People with AS have an increased risk of 60% for cerebrovascular mortality, and an overall increased risk of 50% for vascular mortality. About one third of those with ankylosing spondylitis have severe disease, which reduces life expectancy.

As increased mortality in ankylosing spondylitis is related to disease severity, factors negatively affecting outcomes include:
 * Male sex
 * Plus three of the following in the first two years of disease:
 * Erythrocyte sedimentation rate (ESR) >30 mm/h
 * Unresponsive to NSAIDs
 * Limitation of lumbar spine range of motion
 * Sausage-like fingers or toes
 * Oligoarthritis
 * Onset <16 years old

Gait
The hunched position that often results from complete spinal fusion can have an effect on a person's gait. Increased spinal kyphosis will lead to a forward and downward shift in center of mass (COM). This shift in COM has been shown to be compensated by increased knee flexion and ankle dorsiflexion. The gait of someone with ankylosing spondylitis often has a cautious pattern because they have decreased ability to absorb shock, and they cannot see the horizon.

Epidemiology
Between 0.1% and 0.8% of people are affected. The disease is most common in Northern European countries, and seen least in people of Afro-Caribbean descent. Although the ratio of male to female disease is reportedly 3:1, many rheumatologists believe the number of women with AS is underdiagnosed, as most women tend to experience milder cases of the disease. The majority of people with AS, including 95 per cent of people of European descent with the disease, express the HLA-B27 antigen and high levels of immunoglobulin A (IgA) in the blood. In 2007, a team of researchers discovered two genes that may contribute to the cause of AS: ARTS-1 and IL23R. Together with HLA-B27, these two genes account for roughly 70 percent of the overall number of cases of the disease.

History

Ankylosing spondylitis was distinguished from rheumatoid arthritis by Galen as early as the 2nd century AD. Skeletal evidence of the disease (ossification of joints and entheses primarily of the axial skeleton, known as "bamboo spine") was thought to be found in the skeletal remains of a 5000-year-old Egyptian mummy with evidence of bamboo spine. However, a subsequent report found that this was not the case.

The anatomist and surgeon Realdo Colombo described what could have been the disease in 1559, and the first account of pathologic changes to a skeleton possibly associated with AS was published in 1691 by Bernard Connor. In 1818, Benjamin Brodie became the first physician to document a person believed to have active AS who also had accompanying iritis.

In 1858, David Tucker published a small booklet which clearly described the case of Leonard Trask, who had severe spinal deformity subsequent to AS. In 1833, Trask fell from a horse, exacerbating the condition and resulting in severe deformity. Tucker reported: "It was not until he [Trask] had exercised for some time that he could perform any labor ... [H]is neck and back have continued to curve drawing his head downward on his breast." The account of Trask became the first documented case of AS in the United States, owing to its indisputable description of inflammatory disease characteristics of AS and the hallmark of deforming injury in AS.

In the late nineteenth century, the neurophysiologist Vladimir Bekhterev of Russia in 1893, Adolf Strümpell of Germany in 1897, and Pierre Marie of France in 1898 were the first to give adequate descriptions which permitted an accurate diagnosis of AS prior to severe spinal deformity. For this reason, AS is also known as Bekhterev disease, Bechterew's disease or Marie–Strümpell disease.

The word is from Greek ankylos meaning crooked, curved or rounded, spondylos meaning vertebra, and -itis meaning inflammation.

title: Ankylosing spondylitis


text:
Baron Sandhurst

Baron Sandhurst, of Sandhurst in the County of Berkshire, is a title in the Peerage of the United Kingdom. It was created on 28 March 1871 for the soldier Sir William Mansfield, Commander-in-Chief of India between 1865 and 1870 and Commander-in-Chief of Ireland between 1870 and 1875. He was the grandson of Sir James Mansfield, Solicitor-General and Chief Justice of the Common Pleas. Lord Sandhurst's eldest son, the second Baron, was a Liberal politician and also served as Governor of Bombay. On 1 January 1917 he was created Viscount Sandhurst, of Sandhurst in the County of Berkshire, in the Peerage of the United Kingdom. However, he had no surviving male issue and on his death in 1921 the viscountcy became extinct. He was succeeded in the barony by his younger brother, the third Baron. the title is held by the latter's great-grandson, the sixth Baron, who succeeded his father in 2002. He is a barrister and judge and was elected to a hereditary-peers' seat in the House of Lords in 2021.

Barons Sandhurst (1871)

 * William Rose Mansfield, 1st Baron Sandhurst (1819–1876)
 * William Mansfield, 2nd Baron Sandhurst (1855–1921) (created Viscount Sandhurst in 1917)

Viscounts Sandhurst (1917)

 * William Mansfield, 1st Viscount Sandhurst, 2nd Baron Sandhurst (1855–1921)

Barons Sandhurst (1871; Reverted)
The heir apparent is the present holder's only son, Hon. Edward James Mansfield (b. 1982)
 * John William Mansfield, 3rd Baron Sandhurst (1857–1933)
 * Ralph Sheldon Mansfield, 4th Baron Sandhurst (1892–1964)
 * John Edward Terence Mansfield, 5th Baron Sandhurst (1920–2002)
 * Guy Rhys John Mansfield, 6th Baron Sandhurst (b. 1949)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Baron Sandhurst


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Talk:Magma (algebra)

Closed
I wonder if the set is required to be closed under the operation—it's not clear from the article. For instance, would it be right to call Z a magma under the operation of taking the average of two integers, possibly yielding a non-integer?

Also, I think the opening sentence is misleading:

 * ... a magma is a particularly simple kind of algebraic structure.

That's probably intended to mean that being a magma imposes little structure on a set, but one could also interpret it as: "If I can just show that my set here is a magma, why, then it must have a particularly simple algebraic structure!" Since highly complicated fields exist (and fields are magmas), that conclusion would be false. —Herbee 16:24, 2004 Feb 27 (UTC)

 * By a binary operation on a set M, we mean a function from M×M to M. I think this is clear from the binary operation article. So a magma is necessarily closed. I agree that "simple" may be misleading - perhaps you can think of a better wording. (A field is not a magma; it's two interrelated magmas.) --Zundark 16:37, 27 Feb 2004 (UTC)

 * It's clear from you, Zundark; thanks for explaining. But it's not clear from the binary operation article, which points out that the term is sometimes used for any binary function, i.e. not necessarily from S × S to S.
 * A field not a magma? A field has a binary operation, so it fulfills the definition of a magma, doesn't it? Sure, it has another operation and more stuff, but that doesn't demagmafy it. Otherwise it would sound like "a bicicle doesn't have a wheel because it has two wheels!"
 * I wonder why they call it a "magma"? Next thing you know they'll rename a field to a "volcano", because of the magma in there...
 * —Herbee 22:11, 2004 Feb 27 (UTC)

I've actually made that basic rather than simple, which is certainly POV or worse.

Charles Matthews 16:53, 27 Feb 2004 (UTC)

I'd like to suggest that the adjective "closed" be dropped from the definition entirely. If we define a magma to be a set M equipped with a "binary operation on M ", then there's really no chance of confusion (the key is the preposition "on".) The more general notion of a function M × M to N, where the codomain is a second set N, shouldn't be called a binary operation on M. The older literature may have used this terminology, but it's relatively rare to encounter it in contemporary writing. Moreover, I think it's confusing: closure is important when dealing with subobjects, but for the object itself, the adjective raises more questions than it answers (e.g., "why wouldn't a function M × M to M be closed? what would it even mean not to be?", etc.)

Peterlittig (talk) 20:11, 30 April 2016 (UTC)

Groupoid vs magma
We have this tendency to use weird terminology, and here's a good example. Groupoids are called magmas by almost nobody, basically just Bourbaki and the Magma computer algebra system. The people who actually study them call them groupoids. (The fact that every external link the page gives calls them groupoids should be a hint.) It's unfortunate that groupoid has two meanings, but it's not like we don't have plenty of tools to handle ambiguity for links. I suggest we move this to "groupoid (blah)", for suitable value for "blah".-- Walt Pohl 01:12, 16 Mar 2004 (UTC)

 * The title uses disambiguation anyway, so I don't see any objection to moving it. But I don't know what "blah" should be - "algebra" doesn't seem specific enough, as the other type of groupoid can also be considered as an algebraic object. --Zundark 17:22, 17 Dec 2004 (UTC)

I'd only ever heard of 'magma' - and it hadn't occurred to me that groupoid was ambiguous. Sorry, the other groupoid meaning here would be a horrid addition, except as a redirect. 'Magma' is good enough for Jean-Pierre Serre (I learned it from his lectures on Lie algebras), which makes it good enough for me.

Charles Matthews 17:30, 17 Dec 2004 (UTC)

 * But nobody who actually studies them calls them magmas. They call them groupoids.  Magma is a Bourbaki-ism (so it's not surprising that Serre uses it) that never caught on among the actual practitioners of groupoids.

 * Well, I quibble at that. He's fussy about nomenclature, and not uncritical of Bourbaki. I have never come across a specialist in these things; apart that is from User:Deflog who added all these bitty definitions. But that is enough to undermine the idea that no one who works in the area says magma? Category theorists don't count? Well, it's anyway kind of obvious that category theorists would hate having 'groupoid' here. Charles Matthews

 * As for what to use for disambiguation, I don't know. Maybe 'groupoid (category theory)' for the other definition, and 'groupoid (algebra)' for the magma definition?  I agree that the other kind of groupoids can be considered an algebraic object, so it's not perfect.  Maybe 'groupoid (binary operation)'?  -- Walt Pohl 20:34, 17 Dec 2004 (UTC)

 * I thought this over, and I still think we have no right to dictate which definition of groupoid is the right one. The term is used in two different senses in mathematics, and we should reflect that. -- Walt Pohl 06:32, 25 Jan 2005 (UTC)

 * Two comments:
 * "Magma" is also to me much more common than "groupoid", for this meaning.
 * I'm already worried about the "specification" "(algebra)" and just created a redirect for "Magma (mathematics)". Generalizing the principle that no such "specification" at all is added if there is no ambiguity, it seems most logical to me to add the name of the most largest possible "category" where no ambiguity exists. Else it is virtually impossible to guess for a user, what to add if (s)he searches the definition of an unknown term (and maybe does not even know what the precise meaning is). Thus:
 * Why is this called Magma (algebra) and not Magma (mathematics) ?
 * I really would regret addition of (binary operation) or even less intuitive things to "Magma", and I regret wherever an unnecessarily restrictive qualifier is added. (This seems b.t.w. especially common (almost exclusive) in this area of abstract algebra.)
 * Just think about a term of everyday life or of a domain you're not common with, and what you would do if you were searching for its definition, lacking a disambiguation page. MFH 20:52, 17 Mar 2005 (UTC)

tinyurl.com/uh3t here -- I learned "groupoid" in school, not "magma". Today on WikiPedia was my first sighting of "magma" with this meaning, as contrasted with melted basalt under the Earth's crust. I strongly feel a disambiguation page is needed. If somebody, as I did, searches for "groupoid", he should see *both* definitions, via diaambiguation page, rather than only the category-theory one which is there now. Move the category-theory one qualified name such as groupoid (category theory), and either have brief groupoid (algebra) that points to Magma, or just link to here directly from disambiguation page.

On the Mathematics Subject Classification (2000) the word magma apparently never appears. I believe that there is no doubt that MSC represents standard current mathematical terminology, hence I think that the page should be renamed Groupoid (algebra). Popopp 16:21, 13 February 2007 (UTC)

 * As already pointed out, the problem with using groupoid (algebra) is that the categorical meaning can also be defined as an algebraic structure. So groupoid (algebra) is still ambiguous. -- Fropuff 18:39, 13 February 2007 (UTC)

 * You are right; I was not familiar with the other notion of Groupoid, so I missed the main point that the other notion is algebraic, too. Anyway, I still do not see why the present notion should be called Magma (see also below). Since the main difference between the two notions is that one is a partial algebraic structure and the present one is a total algebraic structure, I believe that the right name for the present one could be Groupoid (total algebraic structure), while the former Groupoid page might be eventually (but not necessarily) moved to Groupoid (partial algebraic structure). One might also consider the number of operations, hence call this one Groupoid (one operation). In order to definitely check the status of the use of magma among mathematics, I have made a search within AMS mathscinet database; I searched the word magma anywhere in reviews of papers with Primary classification 20n02 (Sets with a single binary operation). The search has given two matches, while the search of the word groupoid has given 172 matches! Searching on Arxiv.org is not that easy, I looked at some results for magma-anywhere, and I have found one case in which it is used in the present sense, but I have found many other uses for it (for example,  uses it for still another kind of algebraic structure-four operations!). My conclusion is that Magma is much more unusual and even more confusing than Groupoid (whatever). In my opinion, the only thing to do is to choose the most appropriate whatever. As I mentioned, I suggest Groupoid (total algebraic structure). —The preceding unsigned comment was added by 82.84.201.129 (talk) 21:04, 13 February 2007 (UTC).

I have thougt a lot about the problem. One one side, it is just a matter of terminology, hence it is not very relevant, as far as the mathematical content of the entry is correct. On the other side, it might cause problems to someone wanting to study the notion both on wikipedia and on the mathematical literature, or simply just trying to learn and understand definitions. But what conviced me to talk again about the terminology is the consideration that wikipedia is not original research, and is not original research even in naming conventions. Hence we must adhere to well established conventions which, as I argued, in this case coincide with those used in the Mathematical Subject Classification (and to the use of mathematicians working in the field as well: as I mentioned mathscinet in Primary classification 20n02 gives 2 matches for magma and 172 matches for groupoid!). Because of the above reasons I propose to use Groupoid (Set with a single binary operation), which is the name used in the MSC Classification. Please make me know if you do not agree.--Popopp 08:15, 2 May 2007 (UTC)
 * I do agree. Please make the move ;-) --Tchoř (talk) 11:12, 18 July 2009 (UTC)

Well, this is very confusing for people ignorant of algebra (like me). Could we put a disambiguation in the template: Template:Group-like structures. Because the two books I am reading right now don't give inverses, neutral, etc... to groupoids. Tony (talk) 16:43, 17 September 2010 (UTC)

Whatever happened to moving this page? It sounds like it would be quite appropriate. I'd do it myself, but the most recent suggestion ("Groupoid (Set with a single binary operation)") sounds fairly awkward. However, I don't know ewnough about the topic to suggest a better alternative. I'll try finishing the article and seeing if I can think of something better by then. If someone smarter than me could do it instead or at least offer an alternative, it'd be appreciated. 71.199.190.190 (talk) 00:32, 3 September 2011 (UTC)

The current naming of Wikipedia is the best in practice. There are two very different kinds of objects: (1) small categories where every morphism is invertible and (2) sets with a binary operation. For (1) there is only one well-established name, 'groupoid', and for (2) there are two well-established names, 'magma' and 'groupoid'. So it is just better to avoid collision, and use 'magma' for (2) --- and of course, mention the alternate naming for it. --- 2011-11-27

Division vs cancelation
Would it be better to say that quasigroups allow cancelation, rather than division? Division implies that there is an inverse for each element.

(a*b)*c = (a*c)*(b*c)
In a completely unrealated subject I found that when I expressed a certian process as a binary operater it followed the identity (a*b)*c = (a*c)*(b*c). Is there any significance to this identity? --SurrealWarrior 19:50, 22 January 2006 (UTC)

 * It means that the operation is right distributive over itself. Any binary operation that is both idempotent and medial will have this property. For example, * could be the operation of taking the midpoint of two points in a real vector space. I don't know if this answers your question. --Zundark 20:51, 22 January 2006 (UTC)

 * Thank you! The process I was talking about is quite similar to taking the midpoint of two points in a real vector space, it is mixing two ideas to produce a third, for instance, mixing sports and player to produce a sports player.--SurrealWarrior 21:02, 22 January 2006 (UTC)

 * That sounds like you're looking for quandles. RandomP 19:04, 29 April 2006 (UTC)

Monad
According to [] a magma is synonymous for monad?

In dutch 'monade' is our name for that.

What do you think?

Evilbu 17:38, 16 February 2006 (UTC)
 * I think maybe the term magma isn't so standard. Some people call magmas groupoids (a usage which I don't condone, since groupoids should be categories with all invertible morphisms).  Similarly, I guess some people call magmas monads, like the link you show.  I don't like that usage either, because a monad should be monoid object in a monoidal category.  Groupoids, monoids, and monads have identities and associativity, while magmas do not, so the different definitions are not very related.  Of course, you can use any name you want, but using the name "groupoid" or "monad" for magma strikes me as unnecessarily ambiguous. -lethe talk [ +] 00:34, 17 February 2006 (UTC)

Free magma/binary tree automorphisms endomorphisms
It seems to me that a free magma is necessarily endowed with any operation that can be performed on a (rooted, complete) binary tree that leaves the binary tree rooted and complete. Right? In essence, magmas have well-defined, natural automorphisms endomorphisms. Yet, the discussion here, and corresponding discussion in the cat theory pages are notable in being silent on this matter. Is this ommission due to an incompleteness of the article, or is this due to some unwillingness or "incorrectness" of discussing this topic? Maybe this is an expression of "forgetfullness" in the cat theory sense? In which case, is there a category of "free magmas with their natural automorphisms"? linas 20:23, 11 February 2007 (UTC)

 * It's true that any magma has an automorphism group and that the elements of a free magma $$\mathbf{F}(X)$$ on a generating set $$X$$ can be identified with the rooted binary trees labeled with a member of $$X$$ at each leaf (with the assignment of a «left» and «right» child node for each nonleaf node). I don't recall ever seeing a detailed account of automorphisms of free magmas and a short search just now returned nothing of note. Off the top of my head I'm not seeing any other automorphisms than those induced by permuting the elements of $$X$$, but there may well be other examples. It's worth noting that an automorphism of $$\mathbf{F}(X)$$ permutes the set of those labeled trees, not the nodes of the trees themselves. If you're thinking that automorphisms of the free magmas are «natural» in the sense that they induce automorphisms of all other magmas that is incorrect. There is an automorphism of $$\mathbf{F}(\{x,y\})$$ which swaps $$x$$ and $$y$$, yet the abelian group $$\mathbb{Z}_2\oplus\mathbb{Z}_3$$ (viewed as a magma) is a quotient of $$\mathbf{F}(\{x,y\})$$ generated by $$\{(1,0),(0,1)\}$$ which does not have an automorphism swapping these two generators. That is, automorphisms of free magmas do not descend to automorphisms of all other magmas. The concept of «forgetfulness» in category theory doesn't apply here, since that is more about dropping structure as when we forget about the binary operation on a magma and only look at its underlying set. There is a category whose objects are the free magmas and whose morphisms are all the free magma automorphisms. It is a subcategory of the usual category whose objects are magmas and whose morphisms are magma homomorphisms. caterpillar_tree (talk) 23:17, 20 April 2020 (UTC)

 * The original question was sloppy. The "natural maps" are the ones generated by the left-subtree map $$L: (a b)\mapsto a$$, the right subtree map $$R: (a b)\mapsto b$$, and the hyperbolic rotation $$\theta: ((a b) c)\mapsto (a (b c))$$. Note that $$\theta$$ is invertible. Obviously, these do not commute with the magma product itself; they are endomorphisms only in the sense that given an element of $$\mathbf{F}(X)$$, they return another element of $$\mathbf{F}(X)$$. They do preserve morphisms, in the sense that if $$f:M\to N$$ is a morphism of magmas, these three maps all commute with $$f$$ in that $$f\circ L = L\circ f$$ and $$f\circ \theta = \theta\circ f$$  and etc. Because they commute, the universal property is conserved. Because the universal property is preserved, these are "natural". These are not the only ones, they generated an infinite set: for example, you can make some arbitrary number of left-right moves down the tree, and then apply $$\theta$$ at that location, but keep the rest of the tree above that point unchanged. So there's an infinite number of these maps that are natural. The "rotation" is "hyperbolic" because binary trees embed into the modular group (which describes the hyperbolic rotations of the hyperbolic plane). And once you say these magic words, you've opened the flood-gates to all of riemann surfaces and number theory and what-not. So lets keep those flood-gates closed. Besides these, there are also also the tree-editing maps: e.g. take a bunch of left-right moves down the tree, take whatever you find there, and replace it with a different tree. So these are a raft of "self-evident" operations; seems like they belong in the article, right? 67.198.37.16 (talk) 01:21, 14 October 2020 (UTC)

 * I'm not sure I'm following you here. If $$f\colon\mathbf{M}\to\mathbf{N}$$ is a magma homomorphism then $$f\colon M\to N$$ is a function which eats a single member of $$M$$ at a time. You claim that $$f\circ L=L\circ f$$ but as far as I can see $$L\colon F(X)\to F(X)$$ is a function taking elements of the universe of the free magma (so in other words those rooted binary trees whose leaves are labeled with members of $$X$$) to other elements of the same set, which makes it looks like these two functions $$f$$ and $$L$$ cannot be composed with each other in any sensible way. Do you care to elaborate on what you meant by this? caterpillar_tree (talk) 20:35, 15 December 2021 (UTC)

Division ?
It would be helpful to provide an information about what division is. Following the given link take to a page where the best one can find is the section on abstract algebra, where division is "defined" as multiplication with the inverse. Here it seems you talk about something else. What can it be? The best I can imagine is the possibility to solve uniquely(?) the equation a x=b for any a,b. Must this solution be unique? Or is it only the surjectivity of left-multiplication, for all elements? I think several definitions are possible, but even knowing semigroups, magmas, etc quite well, I believe that this notion is far from being standard and should be explained.— MFH:Talk 21:46, 12 February 2007 (UTC)

Because magma is a relatively simple, intuitive structure, the notions of Galois connection, free functor could be at least mentioned in the Free magma section
Would it be worth to mention the (rather hard) concepts of Galois connection and Adjoint functor using the (intuitively simple) example of magma? Now, these concepts are demonstrated on the example of free group — but I think a free group is a rather hard concept, a free magma is much more familiar and didactic (overall usage and applications in computer science). I thought of the following modifications on section Free magma:
 * 1) A clear distinction between algebraic structure (typeset with fraktur) and set (typeset with Roman), especially carefully distinguishing structure vs its universe (otherwise, the very "flavor" of Galois connection and adjoint functor cannot be rendered well didactically. Even if we do not mention forgetful functor explicitly, but at least the notation could refer to its tacit usage implicitly.
 * 2) Adding at least a summarizing mentioning of these concepts (marked below with red).

Thus, the section Free magma would look like this:

A free magma $$\mathfrak M_X$$ on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object). It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labeled by elements of X. The operation is that of joining trees at the root. It therefore has a foundational role in syntax.

A free magma has the universal property such that, if $$f:X\to N$$ is a function from the set X to the universe N of any magma $$\mathfrak N$$, then there is a unique extension of $$f$$ to a morphism of magmas $$f^\prime$$

 * $$f^\prime: \mathfrak M_X \to \mathfrak N.$$

This property establishes a special relatedness among two sets and two algebraic structures, where N is gotten from $$\mathfrak N$$ by “forgetting” the structure imposed by the operation ( forgetful functor ), and $$\mathfrak M_X$$ is gotten from X by regarding it as generators of a [free] algebraic structure ( free functor ). This special interlinkedness (somewhat resembling to Galois connection ) can be generalized to the notion of  adjoint functor.

See also: free semigroup, free group, Hall set

-- 23:14, 27 February 2008 User:Physis

 * The "somewhat resembling" is problematic. Anything adjoint to something forgetful "somewhat resembles" anything that's adjoint to something else. 67.198.37.16 (talk) 01:41, 14 October 2020 (UTC)

Algebraic structure of NAND and NOR functions?
Is there a specific name for the algebraic structure of the boolean NAND(sheffer) and Logical_NOR(peirce) functions? They satisfy no other "common" algebraic axioms except for being closed, unital and commutative. —Preceding unsigned comment added by 129.13.186.1 (talk) 21:58, 6 August 2008 (UTC)

Are there associative, divisible magmas without identity elements and/or non-associative magmas with identity elements but not inverse elements?
Looking at the "Types of magmas" section of this article, in the Hexagonal-shaped image of two paths of successive additional membership creteria from a magma to a group (divisibility-identity-associativity and associativity-identity-inversibility, inversibility being equivilent to divisibility where there is identity and associativity, and even without associativilty if different left- and right inverses are allowed in the definition of inversibility), I have wondered if a path of proper subsets could be formed either with identity first or identity last (or both). Can a magma have an identity element without being either associative or even left- and right-inversible (divisible)? It would surprise me if a magma could have an indentity element without being associative as long as was divisible and without being divisible as long as it was associative but not if the magma was neither associative nor divisible. And can a magma be associative and divisible (an associative quasigroup or a divisible semigroup, or equivilently both a quasigroup and a semigroup) without also having an identity element and thus being a group? Are there 2 paths from a magma to a group forming a hexagon of subsets or 3! = 6 paths forming a cube of subsets? Finally, are there special names for magmas with identity elements (if they aren't all loops and/or monoids including groups) or quasigroups that are also semigroups (if they aren't all groups)? Thanks. Kevin Lamoreau (talk) 19:40, 6 September 2008 (UTC)

 * "Can a magma have an identity element without being either associative or even left- and right-inversible (divisible)?" Yes: the octonions under mutliplication have an identity element (1), but are neither commutative nor associative.  On the other hand, the octonions do have multiplicative inverses.  Sullivan.t.j (talk) 20:50, 6 September 2008 (UTC)

 * I could have phrased that question better. I knew about monoids like the octonions, put I was wondering, in that question if a magma could satisfy three conditions: (1) existence of an identity element; (2) lack of associtativity; and (3) lack of divisibility (and thus lack of inversability).  My second question was if a magma could satisfy the following three conditions: (1) absence of an identity element; (2) associativity; and (3) divisibility (the opposite of the first case).  My final question was what names there were if there were any for those sets (or rather the ones whose members could satisfy all of those criteria like associativity, identity, divisibility and inversability but only had to satisfy identity in the first case and associativity and divisibility in the second case.  I wasn't concerned with whether or not a magma was commutative here.  Thanks for your attempt to answer part of my question though. Kevin Lamoreau (talk) 00:11, 7 September 2008 (UTC)

 * Your first set of conditions is easily satisfied - fill in a 5x5 multiplication table at random, add an identity element, and you are very likely to have an example. The 3-element magma I posted here appears to be an example too, but all its elements have an inverse (which you seem to think is ruled out by the lack of divisibility - maybe I've misunderstood what you meant). Your second of set of conditions is impossible to satisfy, unless you count the empty semigroup.

 * I don't think there's a special term for a magma with an identity element - just "groupoid with an identity element" would be the usual expression. A magma with associativity and divisibility is called a group (unless it's empty). --Zundark (talk) 08:09, 7 September 2008 (UTC)

Is or is not a magma a groupoid?
In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure.

http://en.wikipedia.org/wiki/Magma_(algebra)#Types_of_magmas

Is or is not a magma a groupoid? In the section "Types of magmas", groupoid completely reverses the definition of a magma. —Preceding unsigned comment added by Fantadox (talk • contribs) 13:01, 30 November 2009 (UTC)

 * If you're referring to the "Group-like structures" table, that table is using the other (category-theoretic) meaning of "groupoid". I'm inclined to remove the table on the grounds that it's needlessly confusing and doesn't really add anything to the article anyway. --Zundark (talk) 13:48, 30 November 2009 (UTC)

"Inversibility"?
Is "inversibility" a word? Is it the one frequently used in this topic? What is wrong with "invertibility" — does it mean something else? Shreevatsa (talk) 05:27, 30 May 2010 (UTC)
 * Note: I have changed the article to say "invertibility" for now, on the assumption that the old word was just a mistake (this article is the only one on Wikipedia that contained the word). If I'm wrong, I'd like to know. Shreevatsa (talk) 05:01, 31 May 2010 (UTC)

a simple solution to the terminology "groupoid" ambiguity is the name 'nonassociative semigroup' or even shorter 'pre-semigroup', where the prefix 'pre-' is an abbreviation for 'nonassociative' Doc at ut (talk) 17:42, 14 March 2011 (UTC)doc at ut

'Semicategory'?
Is 'semicategory' the standard term for a set with a partial associative binary operation? As the name for a set endowed with a total associative binary operation is called a 'semigroup', and when we remove the axiom of identity from the definition of a monoid, we do not call it a 'semimonoid'. Thus, calling its non-closed cousin a 'semicategory' is inconsistent nomenclature. I propose we use the term 'semigroupoid' here and mention any alternate names on the target page. Further supporting this suggestion is, a) the 'semicategory' link actually redirects to the 'semigroupoid' page anyway; b) on a lesser note, 'semigroupoid' is the preferred term for the Haskell implementation of this algebraic structure. MaxwellEdisonPhD (talk) 12:50, 14 September 2013 (UTC)
 * Probably not or at least has multiple defs. See Talk:Semigroupoid. JMP EAX (talk) 11:42, 26 August 2014 (UTC)

Reason
The article says, "Magmas are not often studied as such". The lack of interesting theorems might be the reason. — Preceding unsigned comment added by 64.38.197.205 (talk) 11:02, 25 July 2014 (UTC)

Bourbaki
According to Bourbaki called a magma what today is called a semigroup, i.e. having associativity. The source could be wrong though, but it was written by two French sounding names... JMP EAX (talk) 12:15, 26 August 2014 (UTC)

According to "magma" with the meaning given in this article was used [and probably introduced] by Serre. JMP EAX (talk) 12:32, 26 August 2014 (UTC)

 * Why not go straight to the source? Bourbaki's Algebra I defines a "magma" on page 1 .  It is not required to be associative.  On page 4 is the definition of an "associative magma".  Deltahedron (talk) 16:41, 26 August 2014 (UTC)  Additional.  It happens that I found a copy of the 1951 fascicule of Algebre I.  This starts by defining a partial binary operation ("loi de composition interne") which need not satisfy associativity.  As far as I can tell, the word magma does not appear in this version.  According to  the term "magma" appears in the 1971 edition.  Kiechle  states that Bourbaki first used the term magma in that edition.  Incidentally, the 1951 edition defines as "monoide" what we would now call a semigroup, that is, an associative everywhere defined operation.  This may be what was intended by Gondran & Minoux.  Deltahedron (talk) 18:53, 26 August 2014 (UTC)

Zeropotent?
The article has a definition of a zeropotent magma that is not equivalent to nilpotent. I cannot find this term used at all in a Google search. Should we delete this definition? —Quondum 21:32, 11 January 2015 (UTC)

 * A simple GScholar search shows 46 results. This paper is the first hit. It seems a verifiable term. --Mark viking (talk) 22:44, 11 January 2015 (UTC)

 * I was searching for "magma", where there were no results. I'll ignore references that only define zeropotent in the context of a more restricted structure (e.g. semigroups), such as the article you linked to.  I see that there are several references that use the equivalent term groupoid, though.  From, we get:
 * "By a groupoid we mean a non-empty set with one binary operation. If $G$ is a groupoid then an element o£ $G$ is said to be absorbing if $xo = o = ox$ for every $x ∈ G$. A groupoid $G$ with an absorbing element $o$ is said to be zeropotent if $xx = o$ for every $x ∈ G$ and $G$ is said to be a Z-semigroup if $xy = o$ for all $x,y ∈ G$."
 * So basically, a zeropotent magma is one in which every element squares to the two-sided absorbing element, which must exist. The identities in the article seem to be a poor way to capture that, and indeed seem to me to be nonequivalent (they do not imply a right zero). I'll make changes to make them equivalent. —Quondum 23:32, 11 January 2015 (UTC)

Emptyness
Hi !

I wonder if the definition of a magma requires the set to be non-empty. If so, i think it would be relevent to mention it in the article...

Best regards ! — Preceding unsigned comment added by 109.23.11.115 (talk) 18:54, 8 March 2015 (UTC)

 * This definition in this article does not require the set to be nonempty. Some authors may exclude empty objects from consideration in their definitions, but I suspect that has more to do with not being bothered to think about it. I don't see any change needed in the article in this regard. —Quondum 21:02, 8 March 2015 (UTC)

Group doesn't possess an Identity element?
In the table labeled "Group-like structures", it has groups and the empty domain under the same category, and states that neither groups nor the empty domain require an identity element. Not a big deal, but does seem like an error. The empty domain should be given it's own row perhaps. — Preceding unsigned comment added by 2600:1700:C821:39D0:FD8C:2365:6D39:F3B2 (talk) 20:31, 23 June 2022 (UTC)
 * It was added in this edit Feb 22, 2022, changing from Inverse semigroup, note Inverse semigroups do not belong in the table; what fits, if anything, are possibly-empty groups. I don't know of a better wording, however.. I'll comment out the confusing row for now. Tom Ruen (talk) 21:46, 23 June 2022 (UTC)

Homogeneous element
What is the definition of homogeneous element in free magma? 213.6.145.233 (talk) 09:19, 26 September 2022 (UTC)

title: Talk:Magma (algebra)


text:
Baron Sackville

Baron Sackville, of Knole in the County of Kent, is a title in the Peerage of the United Kingdom. It was created in 1876 for the Honourable Mortimer Sackville-West, with remainder, failing heirs male of his body, to his younger brothers the Hon. Lionel and the Hon. William Edward. Sackville-West was the fourth son of George Sackville-West, 5th Earl De La Warr and Elizabeth Sackville-West, Countess De La Warr and 1st Baroness Buckhurst, younger daughter and co-heir of John Sackville, 3rd Duke of Dorset. On the death of the latter's cousin, Charles Sackville-Germain, 5th Duke of Dorset, in 1845, the dukedom and its subsidiary titles became extinct and the Sackville estates passed through Elizabeth to the West family who assumed the additional surname of Sackville by Royal licence. By arrangement, Mortimer Sackville-West succeeded to a substantial part of the estates, including Knole in Kent, which is still the seat of the Barons Sackville.

He was succeeded in the barony according to the special remainder by his brother Lionel, who became the second Baron. He had no legitimate male issue and, on his death, the title passed to his nephew, the third Baron. He was the son of the aforementioned William Edward. He was succeeded by his younger brother, the fourth Baron. He was a major-general in the army. On his death the title passed to his son, the fifth Baron, and then to the latter's cousin, the sixth Baron. He was the eldest son of the Hon. Bertrand George Sackville-West, youngest brother of the fourth Baron. the title is held by his nephew, the seventh Baron, who succeeded in 2004. He is the son of Hugh Rosslyn Inigo Sackville-West, younger brother of the sixth Baron.

The poet Vita Sackville-West was the daughter of the third Baron and his wife Victoria Sackville-West, daughter of the second Baron.

Barons Sackville (1876)

 * Mortimer Sackville-West, 1st Baron Sackville (1820–1888)
 * Lionel Sackville-West, 2nd Baron Sackville (1827–1908)
 * Lionel Edward Sackville-West, 3rd Baron Sackville (1867–1928)
 * Charles John Sackville-West, 4th Baron Sackville (1870–1962)
 * Edward Charles Sackville-West, 5th Baron Sackville (1901–1965)
 * Lionel Bertrand Sackville-West, 6th Baron Sackville (1913–2004)
 * Robert Bertrand Sackville-West, 7th Baron Sackville (b. 1958)

The heir apparent is the present holder's son Hon. Arthur George Sackville-West (b. 2000).

title: Baron Sackville


text:
Hannibal Sehested (governor)

Hannibal Sehested (1609 – 23 September 1666) was a Dano-Norwegian statesman and son-in-law of King Christian IV. He served as Governor-general of Norway from 1642 to 1651. He fought in the Torstenson War against Sweden and implemented many reforms in Norway. After a fall from grace leading to his resignation as Governor-general in 1651, he regained the trust of Frederick III in 1660 and negotiated the Treaty of Copenhagen. He worked as lord treasurer and councillor of state until his death in 1666.

Early life
Sehested was born at Arensborg Castle on Øsel, in Danish Estonia, as the son of Claus Maltesen Sehested (1558–1612) and Anne Nielsdatter Lykke (1568–1645). He was named after his maternal uncle Hannibal Mogensen Gyldenstjerne of Restrup. He attended the Sorø Academy from 1626 to 1629, and then studied abroad in Germany, Holland, France and England from 1629 to 1639. After completing his education abroad, he returned to Denmark and was attached to the court of King Christian IV. In 1639 he was granted the fiefdom of Tranekær, and in 1640 he received the far more lucrative Båhus fiefdom in Norway and was also appointed member of the Danish National Council.

In 1640 and 1641 he was sent to Wismar to negotiate a treaty with the Swedish chancellor, Axel Oxenstierna, and, if possible, to bring about the marriage of Christian's son Frederick and Gustavus Adolphus's daughter Christina. Though failing in both particulars, he retained the favor of the king, who had marked him out as a son-in-law, one of seven by whose influence he hoped to increase the influence of the crown. Accordingly, in 1636 he was betrothed to one of the daughters, the countess Christine, then aged nine, whom he married in 1642.

In May 1640, Sehested became a member of the Rigsråd. He believed that the proper field for the exercise of his talents was diplomacy, and he openly aspired to be minister of foreign affairs. Despite a successful embassy to Spain in 1640–1641, he did not obtain the coveted post, but was appointed Governor-General of Norway.

Governor-general of Norway under Christian IV
In April 1642 he was appointed Governor-General of Norway, where he served until 1651. He now had the opportunity of displaying his administrative and organizing abilities, united with a remarkable zeal for reform. He made it his main objective to develop Norway's material resources, reorganize the army, fortifications and fiscal system; and he aimed at giving the country a more independent position in the union with Denmark.

The Hannibal War
During Christian IV's second war with Sweden (1643–1645), Sehested, as governor of Norway, assisted his father-in-law materially. He invaded Sweden four times; successfully defended Norway from attack; and, though without any particular military talent, won an engagement at Nysaker in 1644. The war was unpopular with Norway and was referred to in Norway as the Hannibal war. Concerns centered around high taxes required to support the army and the concern that Sweden would be induced to invade Norway. Although Norwegian forces suffered no defeats, as part of the peace settlement, Jemtland and Herjedalen, both Norwegian provinces, were ceded to Sweden by Denmark-Norway with Sehested's acquiescence.

After the war he renewed his reforming efforts, and during the years 1646–1647 strove to withdraw his governorship from the benumbing influence of the central administration at Copenhagen, and succeeded with the help of Christian IV in creating a separate defensive fleet for Norway and giving her partial control of her own finances. He was considerably assisted in his endeavours by the fact that Norway was regarded as the hereditary possession of the Kings of Denmark-Norway.

Fall from favor
At the same time, Sehested freely used his immense wealth and official position to accumulate for himself property and privileges of all sorts. His successes finally excited the envy and disapprobation of the Danish Rigsraad, especially of his rival, Korfits Ulfeldt, also one of the king's sons-in-law. The quarrel became acute when Sehested's semi-independent administration of the finances of Norway infringed upon Ulfeldt's functions as lord treasurer of the whole realm. In November 1647, Ulfeldt carried his point, and a decree was issued that henceforth the Norwegian leaders should send their rents and taxes direct to Copenhagen.

Governor-general of Norway under Frederick III
On the accession of Frederick III (1648) to the throne, Sehested strove hard to win his favor, but an investigation into his accounts as governor conducted by his enemies brought to light such wholesale embezzlement and peculation that he was summoned to appear before a herredag, or assembly of notables in May 1651 to give an account of his whole administration. Unable to meet the charges brought against him, he compromised matters by resigning his governorship and his senatorship, and surrendering all his private property in Norway to the crown.

After the fall
Throughout his trial, Sehested had shown prudence. He gave back three times what he had embezzled. Calculating on the sympathy of Frederick III for a man of his monarchical tendencies, he had nothing to do with the projects of revenge which were the ruin of Korfits Ulfeldt. From 1651 to 1660, he lived abroad. At the end of 1655, he met the exiled Charles II of England at Cologne and lived a part of the following year with him in the Spanish Netherlands. In the summer of 1657, he returned to Denmark, but Frederick III refused to receive him, and he hastily quit Copenhagen. During the crisis of the Second Northern War of 1658, he was at the headquarters of Charles X of Sweden. In seeking the help and protection of the worst enemy of his country, Sehested approached the very verge of treason, but he never quite went beyond it. When, at last, it seemed probable that the war would not result in the annihilation of Denmark-Norway, Sehested strained every nerve to secure his own future by working in the interests of his native land while still residing in Sweden.

The final years
In April 1660, he obtained permission from Frederick III to come to Copenhagen and was finally instructed by him to negotiate with the Swedes. The Treaty of Copenhagen, which saved the honour of Denmark-Norway and brought her repose, was very largely Sehested's work. He was one of the willing abettors of Frederick III during the emergency of 1660, when he re-entered the Danish service as lord treasurer and councilor of state. Both at home and on his frequent foreign missions, he displayed all his old ability. Politically, he played a major role in those years within the queen Sophie Amalie's party, which included men like Jacob Petersen. He was challenged by new rivals like Kristoffer Gabel and his influence seems to have been somewhat fading during his last years but he remained in office until his death. As a diplomat, he in some ways anticipated the views of Peter, count Griffenfeldt, supporting the policy of friendship with Sweden and a French alliance. He died suddenly in Paris, where he was conducting important negotiations. His political testament is perhaps the best testimony to his liberal and statesmanlike views.

Other Sources

 * which in turn cites:
 * Thyra Sehested, Hannibal Sehested (Copenhagen, 1886)
 * Julius Albert Fridericia, Adelsvældens sidste Dage (Copenhagen, 1894)
 * Carl Olaf Bøggild Andersen (1946) Hannibal Sehested: en dansk statsmand (Universitetsforlaget i Århus)

title: Hannibal Sehested (governor)


text:
George Mealmaker

George Mealmaker (10 February 1768 – 30 March 1808) was a Scottish radical organiser and writer, born in Dundee, Scotland. Like his father before him he was a weaver by trade.

Liberty
He was born in the Seagate district of Dundee in 1768 the son of a handloom weaver and became a weaver himself around 1780.

In the late 1780s Mealmaker, along with Thomas Fyshe Palmer formed the Dundee Friends of Liberty group, Mealmaker becoming head of the group from 1796. In 1793 Mealmaker wrote Dundee Address to the Friends of Liberty in which he criticised the tyranny and despotism of the British government, for which Palmer was arrested as being the writer. Despite Mealmaker admitting that it was he, and not Palmer who had written the pamphlet the court found Palmer guilty on the grounds that he had prepared the text for publication and circulated it, and sentenced him to 14 years penal transportation.

Other persons arrested in relations to the political activities of the Edinburgh Society of the Friends of the People and associated groups like The Dundee Friends of Liberty were Thomas Muir, William Skirving, Maurice Margarot and Joseph Gerrald. These five were transported to Australia in 1794 and 1795 and were collectively known as the Scottish Martyrs to Liberty.

Transported
Mealmaker continued his radical activities and became involved in the Society of the United Scotsmen. He continued to write treatises critical of the government and in 1797 he wrote The Moral and Political Catechism of Man for which he was arrested. Tried at the High Court of Justiciary in Edinburgh by David Rae, Lord Eskgrove on 10/11 January 1798 for sedition and found guilty, he was transported for 14 years to Australia.

This ended a short married life with Marjory Thoms (died c. 1843) of Dundee, whom he had married in 1795 and by whom he had two daughters.

Mealmaker's skills were needed in the colony, and he was put in charge of a weaving factory in Parramatta by Governor King. For several years he did well, and was granted a conditional pardon and other rewards. The next Governor, William Bligh, did not place the same value on developing weaving, a fire partly destroyed the factory in 1807, and Mealmaker died destitute in 1808, of alcoholic suffocation.

Outlook
Mealmaker's "political ideology and outlook, which were formed from an amalgam of Paineite republicanism, radical Presbyterianism, and a commitment to religious freedom and democratic parliamentary reform, distanced him from the mainstream of Scottish radicalism of this period."

title: George Mealmaker


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Talk:Tomogram

The most recent edit has actually removed the definition of tomogram.

Charles Matthews 16:33, 27 Feb 2004 (UTC)

I couldn't resist changing this radically. The definition of tomography that was offered invalidates the original usage of tomography, which I admit is almost historical, but we cannot just ignore it. Etymologically, too, tomography means section (i.e. 2D) imaging and not volume reconstruction. I removed the bit about the night sky, because the analogy was completely creaking at the seams. Soundray 16:36, 27 Feb 2004 (UTC)

Charles: the tomography entry is a disambiguation page. Maybe the text from tomogram should be moved to the top of it? Or maybe we should create a "Conventional X-ray tomography" entry and split this one up? I don't know. Soundray 16:39, 27 Feb 2004 (UTC) Oops, sorry, I got you now. The term is now back in. Soundray 16:46, 27 Feb 2004 (UTC)

I made the disambiguation page earlier today - only because tomography was then a redirect to one kind of scan. Perhaps there should now be a merge; I really don't know enough about the area.

Charles Matthews 16:51, 27 Feb 2004 (UTC)

title: Talk:Tomogram


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Baron Strathspey

Baron Strathspey, of Strathspey in the Counties of Inverness and Moray, is a title that has been created twice, both times in the Peerage of the United Kingdom. On both occasions, the barony was created for an Earl of Seafield.

Barons Strathspey, first creation (1858)
It was created for the first time on 14 August 1858 for John Ogilvy-Grant, 7th Earl of Seafield (see the Earl of Seafield for earlier history of this title). This creation became extinct on the death of his son, the eighth Earl and second Baron, in 1884.

Barons Strathspey, second creation (1884)
The barony was revived only a few months after it first became extinct in favour of the late Earl's uncle, James Ogilvy-Grant, 9th Earl of Seafield (who had also succeeded him in the baronetcy of Colquhoun; see Colquhoun baronets for earlier history of this title). The barony and baronetcy remained subsidiary titles of the earldom until the death of the ninth Earl's grandson, the eleventh Earl, in 1915. The earldom, which could be passed on to female heirs, was inherited by the Earl's daughter and only child, the twelfth Countess (see the Earl of Seafield for later history of the earldom).

The baronetcy and barony of Strathspey, which could only be inherited by males, were passed on to the Earl's brother, the fourth Baron. His son, the fifth Baron, was confirmed in the surname of Grant of Grant by decree of the Lord Lyon in 1950. the titles are held by the latter's younger son, the seventh Baron, who succeeded in 2023. His is the 34th Chief of Clan Grant.

Barons Strathspey (1858)

 * John Charles Ogilvy-Grant, 7th Earl of Seafield, 1st Baron Strathspey (1815–1881)
 * Ian Charles Ogilvy-Grant, 8th Earl of Seafield, 2nd Baron Strathspey (1851–1884)

Barons Strathspey (1884)

 * James Ogilvy-Grant, 9th Earl of Seafield, 1st Baron Strathspey (1817–1888)
 * Francis William Ogilvy-Grant, 10th Earl of Seafield, 2nd Baron Strathspey (1847–1888)
 * James Ogilvy-Grant, 11th Earl of Seafield, 3rd Baron Strathspey (1876–1915)
 * Trevor Ogilvy-Grant, 4th Baron Strathspey (1879–1948)
 * Donald Patrick Trevor Grant of Grant, 5th Baron Strathspey (1912–1992)
 * James Patrick Trevor Grant of Grant, 6th Baron Strathspey (1943–2023)
 * Michael Patrick Grant, 7th Baron Strathspey (born 1953)

There is no heir to the barony.

title: Baron Strathspey


text:
Wikipedia:WikiProject Mammals

Welcome to WikiProject Mammals, a group dedicated to improving Wikipedia's coverage of topics related to mammals. If you identify as a mammal and would like to help, please see below and inquire on the talk page.

 * Our goals:
 * To improve Wikipedia's coverage of mammals by creating, expanding, and maintaining articles.
 * To serve as the central point of mammals-related article discussion.
 * To provide guidelines and recommendations for articles about mammals.

 * Our scope:
 * WikiProject Mammals focuses on all mammals, whether common or obscure, living or extinct.

 * What can you do?
 * Add your name to the list of Participants and get started!
 * Look at the navigation box on the right and see what needs doing in the "Tasks" List.
 * Join the Assessment Department and help review article standards and quality.
 * Add this userbox to your user page User WikiProject Mammals
 * Join other Projects like WikiProject Animals

Subprojects and work group

 * WikiProject Mammals/Bats Task Force
 * WikiProject Cats
 * WikiProject Cetaceans (inactive)
 * WikiProject Dogs
 * WikiProject Equine
 * WikiProject Monotremes and Marsupials (inactive)
 * WikiProject Primates
 * WikiProject Rodents (semi-active)
 * WikiProject Mammals/Weasels work group (inactive)

Departments

 * Assessment
 * Collaboration
 * Most Popular pages

Guidelines
The Third edition of Wilson & Reeder's Mammal Species of the World (2005) is the basis for the taxonomic framework used by WikiProject Mammals. The MSW3 classification may be overridden by the conclusions of more recent studies which are widely accepted in secondary sources. MSW3 listings can be found online at the Bucknell University here. (Note: there are some differences between the print edition and the online database. For instance, the online database shows three subspecies of dhole whereas the print edition lists seven. )

The Mammal Diversity Database is a new resource that reflects taxonomic changes since the publication of MSW3. It is accepted practice to override the MSW3 taxonomy if both the Mammal Diversity Database and IUCN agree on the change.

title: Wikipedia:WikiProject Mammals


text:
Talk:Robin (character)

separate pages
I agree also. Other heroes with various incarnations get seperate pages. So should the Robins. Especially Dick, being that he has come into his own. Giant89

I agree with dr bats

Should each of the Robins get their own entry? Tim Drake would get the main entry, Dick would get 'Nightwing', Jason Todd could be just 'Jason Todd,' and there's already a Spoiler entry for Stephanie.

Thoughts? --DrBat 14:18, May 22, 2005 (UTC)

I went ahead and split Grayson and Drake into their own articles.--Kross 09:00, August 30, 2005 (UTC)

Spoiler Warning
I'll go ahead and add a spoiler warning before the section, as it explicitly refers to the death of Jason Todd, and other major plot elements.

All Star Robin
All star batman and robin is only on issue #3 and dick hasn't even appeared as robin yet. i don't think this version needs his own page. at least, not yet. Carrie Kelly doesn't even have her own page and i think she (currently) has more signifigance that All-Star Robin.

''== Robin 3000 ==

Where is the haired, Gun using, Robin of the Future? i kwow who is the wife of robin --110.55.3.5 (talk)--~

Why the Revert War
After my initial confusion regarding the revert-battles going on with the JSA & JLA catagory adds/deletes, I think I've struck upon the reason for it; Earth-2 Robin. I know at least he was in the JSA. Of course, since the continuity Robin isn't nor has he ever been, I certainly don't think that catagorization is correct, but I wonder if it is at the root of this? --mordicai. 16:14, 7 October 2006 (TAE)

 * That's not the issue. A user is trying to imply categories which are not representitive of the mainstream Robin, yes. However, they are also doing it without making any attempt discuss the matter or taking note of a recent ruling on categorization for articles about a legacy. Basically, it's effective vandalism and if they don't stop I'll have to report them. ACS (Wikipedian); Talk to the Ace. See what I've edited. 16:31, 7 October 2006 (TAE)

TAE KAYU :P

Dick Grayson's Boy Wonder place in new continuity
It is of some note that in the post-crisis/zero-hour/infinite crisis continuity there are very (very very) few stories featuring the adventures of Bruce Wayne and Dick Grayson(as Batman and Robin).

In fact it is interesting to note that Robin (the original Boy Wonder - Dick Grayson) was markedly absent from 'Batman: Lengends of the Dark Night' - a publication which was intended to fill in the blanks of Batman's post original crisis history. So far Grayson's adventures of Robin have also not featured in Batman Confidential (though this comic is still in its early days). And 'All Star Batman and Robin' is markedly out of continuity.

In fact, one could probably easily name the in-continuity stories which show how the relationship between Bruce and Dick developed in the early years (in both their private lives and as crime-fighters): Dark Justice & the 'Robin: Year One' Robin annual + one or two others.

I'm not sure how such a point could be made within this article. But I'm sure there is a point somewhere. —The preceding unsigned comment was added by Special:Contributions/ (talk)

Merge proposal
I propose merging the list Robin in other media into the main article Robin as a new section. With some minor adjustments, the content can be seamlessly integrated into the main article. This merge will enhance the context of Robin's character and should not present any issues regarding article size or undue weight. Lililolol (talk) 20:00, 1 June 2024 (UTC)

title: Talk:Robin (character)


text:
Baron St Levan

Baron St Levan, of St Michael's Mount in the County of Cornwall, is a title in the Peerage of the United Kingdom. It was created on 4 July 1887 for the former Member of Parliament Sir John St Aubyn, 2nd Baronet, becoming John St Aubyn, 1st Baron St Levan. He had previously represented Cornwall West in House of Commons as a Liberal and St Ives as a Liberal Unionist. He was succeeded by his eldest son, the second Baron, who was a Colonel and Honorary Brigadier-General in the Grenadier Guards. On his death the titles passed to his nephew, the third Baron, the son of the Hon. Sir Arthur James Dudley Stuart St Aubyn (1867–1897), second son of the first Baron. The third baron was succeeded in 1978 by his eldest son, the fourth baron, who had served with the Royal Navy at Dunkirk and in a minesweeper in Arctic Convoys during World War II and was awarded the Distinguished Service Cross (DSC). As of 2014, the titles are held by the fourth Baron's nephew, the fifth Baron, who succeeded in 2013.

The St Aubyn Baronetcy, of St Michael's Mount in the County of Cornwall, was created in the Baronetage of the United Kingdom in 1866 for the first Baron's father, Edward St Aubyn. He was the illegitimate son of Sir John St Aubyn, 5th Baronet, of Clowance, on whose death in 1839 the St Aubyn baronetcy of Clowance, created in the Baronetage of England in 1671, had become extinct (see St Aubyn Baronets).

The family seat is St Michael's Mount, Cornwall.

St Aubyn Baronets, of St Michaels Mount (1866)

 * Sir Edward St Aubyn, 1st Baronet (1799–1872)
 * Sir John St Aubyn, 2nd Baronet (elevated to the peerage as the 1st Baron St Levan, created in 1887)

Barons St Levan (1887)

 * John St Aubyn, 1st Baron St Levan (1829–1908)
 * John Townshend St Aubyn, 2nd Baron St Levan (1857–1940)
 * Francis Cecil St Aubyn, 3rd Baron St Levan (1895–1978)
 * John Francis Arthur St Aubyn, 4th Baron St Levan, DSC (1919–2013)
 * James Piers Southwell St Aubyn, 5th Baron St Levan (born 1950)

The heir apparent is the present holder's son Hon. Hugh James St Aubyn (born 1983).

title: Baron St Levan