# Differences

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shells [2010/07/29 15:46] (current)
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+=====Shells=====
+
+====Definition====
+A \emph{shell} is a structure $\mathbf{S}=\langle S,+,0,\cdot,1 +\rangle$ of type $\langle 2,0,2,0\rangle$ such that
+
+
+$0$ is an identity for $+$:  $0+x=x$, $x+0=x$
+
+
+$1$ is an identity for $\cdot$:  $1\cdot x=x$, $x\cdot 1=x$
+
+
+$0$ is a zero for $\cdot$:  $0\cdot x=0$, $x\cdot 0=0$
+
+==Morphisms==
+Let $\mathbf{S}$ and $\mathbf{T}$ be shells. A morphism from $\mathbf{S}$
+to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:
+
+$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$, $h(1)=1$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+====Properties====
+^[[Classtype]]  |variety |
+^[[Equational theory]]  |decidable |
+^[[Quasiequational theory]]  | |
+^[[First-order theory]]  |undecidable |
+^[[Locally finite]]  |no |
+^[[Residual size]]  |unbounded |
+^[[Congruence distributive]]  |no |
+^[[Congruence modular]]  |no |
+^[[Congruence n-permutable]]  |no |
+^[[Congruence regular]]  |no |
+^[[Congruence uniform]]  |no |
+^[[Congruence extension property]]  |no |
+^[[Definable principal congruences]]  |no |
+^[[Equationally def. pr. cong.]]  |no |
+^[[Amalgamation property]]  |yes |
+^[[Strong amalgamation property]]  |yes |
+^[[Epimorphisms are surjective]]  | |
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &\\ +f(3)= &\\ +f(4)= &\\ +f(5)= &\\ +f(6)= &\\ +\end{array}$
+
+====Subclasses====
+[[Semirings with identity and zero]]
+
+====Superclasses====
+
+====References====
+
+[(Ln19xx>
+)]

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