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Name  Description 

Allsided Ideal  A nonempty language $L \subseteq \Sigma^*$ is a allsided ideal if $L = \Sigma^*\shuffle L$; the complement is subword converseclosed. 
Bifixclosed  If a language is both prefixclosed and suffixclosed then it is also bifixclosed.
This language is: Prefixclosed $\cap$ Suffixclosed 
Bifixconvex  If a language is both prefixconvex and suffixconvex then it is also bifixconvex. This language is: Prefixconvex $\cap$ Suffixconvex 
Bifixfree 

Factorclosed 

Factorconvex  If $u$, $v$, $w$, $x \in \Sigma^*$ and $w=uxv$ then $x$ is a factor of $w$. A language $L$ is factorconvex if $u$, $v$, $w \in \Sigma^*$, $u$ is a factor of $v$ and $v$ is factor of $w$ with $u$, $w \in L$, then $v \in L$.

Factorfree 

Factorfree regular 
This language is: Factorfree $\cap$ Regular 
Finite  Finite languages are an important subset of regular languages. They are accepted by complete DFAs that are acyclic apart from a loop on the sink state for all alphabetic symbols. 
Left Ideal  A nonempty language $L \subseteq \Sigma^*$ is a left ideal if $L = \Sigma^*L$; the complement is suffix converseclosed. 
Prefixclosed 

Prefixclosed regular 
This language is: Prefixclosed $\cap$ Regular 
Prefixconvex  If $u,v,w \in \Sigma^\star$ and $w = uv$ then $u$ is prefix of $w$. A language $L$ is prefixconvex if $\forall_{ u,v,w }$ $u$ is prefix of $v$ and $v$ is prefix of $w$ with $u,w \in L$ implies $v \in L$. 
Prefixfree 

Prefixfree regular 
This language is: Prefixfree $\cap$ Regular 
Regular  The set of regular languages over an alphabet $\Sigma$ can be defined recursively as follows:
To prove that a language is regular, one can define a DFA, NFA, regular expression, etc. that recognizes that language. 
Right Ideal  A nonempty language $L \subseteq \Sigma^*$ is a right ideal if $L = L\Sigma^*$; the complement is prefix converseclosed. 
Starfree  Starfree languages are the languages constructed from finite languages using only boolean operations and concatenation. 
Subwordclosed 

Subwordconvex  If $w=u_0 v_1 u_1 ... v_n u_n$, where $u_i$ ,$v_i\in\Sigma^*$, then $v=v_1 v_2 ... v_n$ is a subword of $w$.
A language $L$ is subwordconvex if $u$, $v$, $w \in \Sigma^*$, $u$ is a subword of $v$ and $v$ is subword of $w$ with $u$, $w \in L$, then $v \in L$. 
Subwordfree 

Suffixclosed 

Suffixconvex  If $u,v,w \in \Sigma^\star$ and $w = uv$, then $v$ is suffix of $w$. A language $L$ is suffixconvex if $u,v,w \in \Sigma^\star$, $u$ is suffix of $v$ and $v$ is suffix of $w$ with $u,w \in L$ implies $v \in L$. 
Suffixfree 

Suffixfree regular 
This language is: Suffixfree $\cap$ Regular 
Twosided Ideal  A nonempty language $L \subseteq \Sigma^*$ is a twosided ideal if $L = \Sigma^*L\Sigma^*$; the complement is bifix converseclosed. 